Results of Hypergeometric Function

DLMF	Formula	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
15.1.E1	${\displaystyle{\displaystyle{{}_{2}F_{1}}\left(a,b;c;z\right)=F\left(a,b;c;z\right)}}$	hypergeom([a , b], [c], z)= hypergeom([a, b], [c], z)	HypergeometricPFQ[{a , b}, {c}, z]= Hypergeometric2F1[a, b, c, z]	Successful	Successful	-	-
15.1.E1	${\displaystyle{\displaystyle F\left(a,b;c;z\right)=F\left({a,b\atop c};z\right)}}$	hypergeom([a, b], [c], z)= hypergeom([a, b], [c], z)	Hypergeometric2F1[a, b, c, z]= Hypergeometric2F1[a, b, c, z]	Successful	Successful	-	-
15.1.E2	${\displaystyle{\displaystyle\frac{F\left(a,b;c;z\right)}{\Gamma\left(c\right)}=\mathbf{F}\left(a,b;c;z\right)}}$	(hypergeom([a, b], [c], z))/(GAMMA(c))= hypergeom([a, b], [c], z)/GAMMA(c)	Divide[Hypergeometric2F1[a, b, c, z],Gamma[c]]= Hypergeometric2F1Regularized[a, b, c, z]	Successful	Successful	-	-
15.1.E2	${\displaystyle{\displaystyle\mathbf{F}\left(a,b;c;z\right)=\mathbf{F}\left({a,b\atop c};z\right)}}$	hypergeom([a, b], [c], z)/GAMMA(c)= hypergeom([a, b], [c], z)/GAMMA(c)	Hypergeometric2F1Regularized[a, b, c, z]= Hypergeometric2F1Regularized[a, b, c, z]	Successful	Successful	-	-
15.1.E2	${\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop c};z\right)={{}_{2}{\mathbf{F}}_{1}}\left(a,b;c;z\right)}}$	hypergeom([a, b], [c], z)/GAMMA(c)= hypergeom([a , b], [c], z)	Hypergeometric2F1Regularized[a, b, c, z]= HypergeometricPFQRegularized[{a , b}, {c}, z]	Failure	Successful	Skip	-
15.2.E1	${\displaystyle{\displaystyle F\left(a,b;c;z\right)=\sum_{s=0}^{\infty}\frac{{\left(a\right)_{s}}{\left(b\right)_{s}}}{{\left(c\right)_{s}}s!}z^{s}}}$	hypergeom([a, b], [c], z)= sum((pochhammer(a, s)*pochhammer(b, s))/(pochhammer(c, s)*factorial(s))*(z)^(s), s = 0..infinity)	Hypergeometric2F1[a, b, c, z]= Sum[Divide[Pochhammer[a, s]*Pochhammer[b, s],Pochhammer[c, s]*(s)!]*(z)^(s), {s, 0, Infinity}]	Failure	Successful	Skip	-
15.2.E2	${\displaystyle{\displaystyle\mathbf{F}\left(a,b;c;z\right)=\sum_{s=0}^{\infty}\frac{{\left(a\right)_{s}}{\left(b\right)_{s}}}{\Gamma\left(c+s\right)s!}z^{s}}}$	hypergeom([a, b], [c], z)/GAMMA(c)= sum((pochhammer(a, s)*pochhammer(b, s))/(GAMMA(c + s)*factorial(s))*(z)^(s), s = 0..infinity)	Hypergeometric2F1Regularized[a, b, c, z]= Sum[Divide[Pochhammer[a, s]*Pochhammer[b, s],Gamma[c + s]*(s)!]*(z)^(s), {s, 0, Infinity}]	Successful	Failure	-	Skip
15.2.E3	${\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop c};x+\mathrm{i}0\right)-\mathbf{F}\left({a,b\atop c};x-\mathrm{i}0\right)=\frac{2\pi\mathrm{i}}{\Gamma\left(a\right)\Gamma\left(b\right)}(x-1)^{c-a-b}\mathbf{F}\left({c-a,c-b\atop c-a-b+1};1-x\right)}}$	hypergeom([a, b], [c], x + I*0)/GAMMA(c)- hypergeom([a, b], [c], x - I*0)/GAMMA(c)=(2*Pi*I)/(GAMMA(a)*GAMMA(b))*(x - 1)^(c - a - b)* hypergeom([c - a, c - b], [c - a - b + 1], 1 - x)/GAMMA(c - a - b + 1)	Hypergeometric2F1Regularized[a, b, c, x + I*0]- Hypergeometric2F1Regularized[a, b, c, x - I*0]=Divide[2*Pi*I,Gamma[a]*Gamma[b]]*(x - 1)^(c - a - b)* Hypergeometric2F1Regularized[c - a, c - b, c - a - b + 1, 1 - x]	Failure	Failure	Fail -487.8169477+316.7970546*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), x = 3/2} -50479.20446+110828.3499*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = 2^(1/2)-I*2^(1/2), x = 3/2} -163338913.7+140350694.8*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = -2^(1/2)-I*2^(1/2), x = 3/2} 45522.85325+151628.5675*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = -2^(1/2)+I*2^(1/2), x = 3/2} ... skip entries to safe data	Fail Complex[-487.81694810081785, 316.7970557265091] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, Rational[3, 2]]} Complex[-50479.2047214623, 110828.3501851795] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[x, Rational[3, 2]]} Complex[-1.633389146290397*^8, 1.4035069565108505*^8] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[x, Rational[3, 2]]} Complex[45522.853876442554, 151628.56778915678] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[x, Rational[3, 2]]} ... skip entries to safe data
15.2#Ex1	${\displaystyle{\displaystyle\lim_{c\to-n}\frac{F\left(a,b;c;z\right)}{\Gamma\left(c\right)}=\mathbf{F}\left(a,b;-n;z\right)}}$	limit((hypergeom([a, b], [c], z))/(GAMMA(c)), c = - n)= hypergeom([a, b], [- n], z)/GAMMA(- n)	Limit[Divide[Hypergeometric2F1[a, b, c, z],Gamma[c]], c -> - n]= Hypergeometric2F1Regularized[a, b, - n, z]	Successful	Successful	-	-
15.2#Ex1	${\displaystyle{\displaystyle\mathbf{F}\left(a,b;-n;z\right)=\frac{{\left(a\right)_{n+1}}{\left(b\right)_{n+1}}}{(n+1)!}z^{n+1}F\left(a+n+1,b+n+1;n+2;z\right)}}$	hypergeom([a, b], [- n], z)/GAMMA(- n)=(pochhammer(a, n + 1)*pochhammer(b, n + 1))/(factorial(n + 1))*(z)^(n + 1)* hypergeom([a + n + 1, b + n + 1], [n + 2], z)	Hypergeometric2F1Regularized[a, b, - n, z]=Divide[Pochhammer[a, n + 1]*Pochhammer[b, n + 1],(n + 1)!]*(z)^(n + 1)* Hypergeometric2F1[a + n + 1, b + n + 1, n + 2, z]	Failure	Failure	Fail Float(undefined)+Float(undefined)*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 1} Float(undefined)+Float(undefined)*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 2} Float(undefined)+Float(undefined)*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 3} Float(undefined)+Float(undefined)*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), n = 1} ... skip entries to safe data	Skip
15.2.E4	${\displaystyle{\displaystyle F\left(-m,b;c;z\right)=\sum_{n=0}^{m}\frac{{\left(-m\right)_{n}}{\left(b\right)_{n}}}{{\left(c\right)_{n}}{n!}}z^{n}}}$	hypergeom([- m, b], [c], z)= sum((pochhammer(- m, n)*pochhammer(b, n))/(pochhammer(c, n)*factorial(n))*(z)^(n), n = 0..m)	Hypergeometric2F1[- m, b, c, z]= Sum[Divide[Pochhammer[- m, n]*Pochhammer[b, n],Pochhammer[c, n]*(n)!]*(z)^(n), {n, 0, m}]	Successful	Successful	-	-
15.2.E4	${\displaystyle{\displaystyle\sum_{n=0}^{m}\frac{{\left(-m\right)_{n}}{\left(b\right)_{n}}}{{\left(c\right)_{n}}{n!}}z^{n}=\sum_{n=0}^{m}(-1)^{n}\genfrac{(}{)}{0.0pt}{}{m}{n}\frac{{\left(b\right)_{n}}}{{\left(c\right)_{n}}}z^{n}}}$	sum((pochhammer(- m, n)*pochhammer(b, n))/(pochhammer(c, n)*factorial(n))*(z)^(n), n = 0..m)= sum((- 1)^(n)*binomial(m,n)*(pochhammer(b, n))/(pochhammer(c, n))*(z)^(n), n = 0..m)	Sum[Divide[Pochhammer[- m, n]*Pochhammer[b, n],Pochhammer[c, n]*(n)!]*(z)^(n), {n, 0, m}]= Sum[(- 1)^(n)*Binomial[m,n]*Divide[Pochhammer[b, n],Pochhammer[c, n]]*(z)^(n), {n, 0, m}]	Successful	Successful	-	-
15.2.E5	${\displaystyle{\displaystyle F\left({-m,b\atop-m-\ell};z\right)=\lim_{c\to-m-\ell}\left(\lim_{a\to-m}F\left({a,b\atop c};z\right)\right)}}$	hypergeom([- m, b], [- m - ell], z)= limit(limit(hypergeom([a, b], [c], z), a = - m), c = - m - ell)	Hypergeometric2F1[- m, b, - m - \[ScriptL], z]= Limit[Limit[Hypergeometric2F1[a, b, c, z], a -> - m], c -> - m - \[ScriptL]]	Failure	Successful	Skip	-
15.2.E6	${\displaystyle{\displaystyle F\left({-m,b\atop-m-\ell};z\right)=\lim_{a\to-m}F\left({a,b\atop a-\ell};z\right)}}$	hypergeom([- m, b], [- m - ell], z)= limit(hypergeom([a, b], [a - ell], z), a = - m)	Hypergeometric2F1[- m, b, - m - \[ScriptL], z]= Limit[Hypergeometric2F1[a, b, a - \[ScriptL], z], a -> - m]	Successful	Successful	-	-
15.4.E1	${\displaystyle{\displaystyle F\left(1,1;2;z\right)=-z^{-1}\ln\left(1-z\right)}}$	hypergeom([1, 1], [2], z)= - (z)^(- 1)* ln(1 - z)	Hypergeometric2F1[1, 1, 2, z]= - (z)^(- 1)* Log[1 - z]	Successful	Successful	-	-
15.4.E2	${\displaystyle{\displaystyle F\left(\tfrac{1}{2},1;\tfrac{3}{2};z^{2}\right)=\frac{1}{2z}\ln\left(\frac{1+z}{1-z}\right)}}$	hypergeom([(1)/(2), 1], [(3)/(2)], (z)^(2))=(1)/(2*z)*ln((1 + z)/(1 - z))	Hypergeometric2F1[Divide[1,2], 1, Divide[3,2], (z)^(2)]=Divide[1,2*z]*Log[Divide[1 + z,1 - z]]	Successful	Failure	-	Successful
15.4.E3	${\displaystyle{\displaystyle F\left(\tfrac{1}{2},1;\tfrac{3}{2};-z^{2}\right)=z^{-1}\operatorname{arctan}z}}$	hypergeom([(1)/(2), 1], [(3)/(2)], - (z)^(2))= (z)^(- 1)* arctan(z)	Hypergeometric2F1[Divide[1,2], 1, Divide[3,2], - (z)^(2)]= (z)^(- 1)* ArcTan[z]	Failure	Successful	Successful	-
15.4.E4	${\displaystyle{\displaystyle F\left(\tfrac{1}{2},\tfrac{1}{2};\tfrac{3}{2};z^{2}\right)=z^{-1}\operatorname{arcsin}z}}$	hypergeom([(1)/(2), (1)/(2)], [(3)/(2)], (z)^(2))= (z)^(- 1)* arcsin(z)	Hypergeometric2F1[Divide[1,2], Divide[1,2], Divide[3,2], (z)^(2)]= (z)^(- 1)* ArcSin[z]	Successful	Successful	-	-
15.4.E5	${\displaystyle{\displaystyle F\left(\tfrac{1}{2},\tfrac{1}{2};\tfrac{3}{2};-z^{2}\right)=z^{-1}\ln\left(z+\sqrt{1+z^{2}}\right)}}$	hypergeom([(1)/(2), (1)/(2)], [(3)/(2)], - (z)^(2))= (z)^(- 1)* ln(z +sqrt(1 + (z)^(2)))	Hypergeometric2F1[Divide[1,2], Divide[1,2], Divide[3,2], - (z)^(2)]= (z)^(- 1)* Log[z +Sqrt[1 + (z)^(2)]]	Failure	Successful	Successful	-
15.4#Ex1	${\displaystyle{\displaystyle F\left(a,b;a;z\right)=(1-z)^{-b}}}$	hypergeom([a, b], [a], z)=(1 - z)^(- b)	Hypergeometric2F1[a, b, a, z]=(1 - z)^(- b)	Successful	Successful	-	-
15.4#Ex2	${\displaystyle{\displaystyle F\left(a,b;b;z\right)=(1-z)^{-a}}}$	hypergeom([a, b], [b], z)=(1 - z)^(- a)	Hypergeometric2F1[a, b, b, z]=(1 - z)^(- a)	Successful	Successful	-	-
15.4.E7	${\displaystyle{\displaystyle F\left(a,\tfrac{1}{2}+a;\tfrac{1}{2};z^{2}\right)=\tfrac{1}{2}\left((1+z)^{-2a}+(1-z)^{-2a}\right)}}$	hypergeom([a, (1)/(2)+ a], [(1)/(2)], (z)^(2))=(1)/(2)*((1 + z)^(- 2*a)+(1 - z)^(- 2*a))	Hypergeometric2F1[a, Divide[1,2]+ a, Divide[1,2], (z)^(2)]=Divide[1,2]*((1 + z)^(- 2*a)+(1 - z)^(- 2*a))	Successful	Successful	-	-
15.4.E8	${\displaystyle{\displaystyle F\left(a,\tfrac{1}{2}+a;\tfrac{1}{2};-{\tan^{2}}z\right)=(\cos z)^{2a}\cos\left(2az\right)}}$	hypergeom([a, (1)/(2)+ a], [(1)/(2)], - (tan(z))^(2))=(cos(z))^(2*a)* cos(2*a*z)	Hypergeometric2F1[a, Divide[1,2]+ a, Divide[1,2], - (Tan[z])^(2)]=(Cos[z])^(2*a)* Cos[2*a*z]	Failure	Failure	Fail .62e-2-.88e-2*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)} .62e-2-.88e-2*I <- {a = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)} .62e-2+.88e-2*I <- {a = 2^(1/2)-I*2^(1/2), z = 2^(1/2)-I*2^(1/2)} .62e-2+.88e-2*I <- {a = 2^(1/2)-I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}	Successful
15.4.E9	${\displaystyle{\displaystyle F\left(a,\tfrac{1}{2}+a;\tfrac{3}{2};z^{2}\right)=\frac{1}{(2-4a)z}\left((1+z)^{1-2a}-(1-z)^{1-2a}\right)}}$	hypergeom([a, (1)/(2)+ a], [(3)/(2)], (z)^(2))=(1)/((2 - 4*a)* z)*((1 + z)^(1 - 2*a)-(1 - z)^(1 - 2*a))	Hypergeometric2F1[a, Divide[1,2]+ a, Divide[3,2], (z)^(2)]=Divide[1,(2 - 4*a)* z]*((1 + z)^(1 - 2*a)-(1 - z)^(1 - 2*a))	Successful	Successful	-	-
15.4.E10	${\displaystyle{\displaystyle F\left(a,\tfrac{1}{2}+a;\tfrac{3}{2};-{\tan^{2}}z\right)=(\cos z)^{2a}\frac{\sin\left((1-2a)z\right)}{(1-2a)\sin z}}}$	hypergeom([a, (1)/(2)+ a], [(3)/(2)], - (tan(z))^(2))=(cos(z))^(2*a)*(sin((1 - 2*a)* z))/((1 - 2*a)* sin(z))	Hypergeometric2F1[a, Divide[1,2]+ a, Divide[3,2], - (Tan[z])^(2)]=(Cos[z])^(2*a)*Divide[Sin[(1 - 2*a)* z],(1 - 2*a)* Sin[z]]	Failure	Failure	Successful	Successful
15.4.E11	${\displaystyle{\displaystyle F\left(-a,a;\tfrac{1}{2};-z^{2}\right)=\tfrac{1}{2}\left(\left(\sqrt{1+z^{2}}+z\right)^{2a}+\left(\sqrt{1+z^{2}}-z\right)^{2a}\right)}}$	hypergeom([- a, a], [(1)/(2)], - (z)^(2))=(1)/(2)*((sqrt(1 + (z)^(2))+ z)^(2*a)+(sqrt(1 + (z)^(2))- z)^(2*a))	Hypergeometric2F1[- a, a, Divide[1,2], - (z)^(2)]=Divide[1,2]*((Sqrt[1 + (z)^(2)]+ z)^(2*a)+(Sqrt[1 + (z)^(2)]- z)^(2*a))	Failure	Failure	Successful	Successful
15.4.E12	${\displaystyle{\displaystyle F\left(-a,a;\tfrac{1}{2};{\sin^{2}}z\right)=\cos\left(2az\right)}}$	hypergeom([- a, a], [(1)/(2)], (sin(z))^(2))= cos(2*a*z)	Hypergeometric2F1[- a, a, Divide[1,2], (Sin[z])^(2)]= Cos[2*a*z]	Failure	Failure	Successful	Successful
15.4.E13	${\displaystyle{\displaystyle F\left(a,1-a;\tfrac{1}{2};-z^{2}\right)=\frac{1}{2\sqrt{1+z^{2}}}\left(\left(\sqrt{1+z^{2}}+z\right)^{2a-1}+\left(\sqrt{1+z^{2}}-z\right)^{2a-1}\right)}}$	hypergeom([a, 1 - a], [(1)/(2)], - (z)^(2))=(1)/(2*sqrt(1 + (z)^(2)))*((sqrt(1 + (z)^(2))+ z)^(2*a - 1)+(sqrt(1 + (z)^(2))- z)^(2*a - 1))	Hypergeometric2F1[a, 1 - a, Divide[1,2], - (z)^(2)]=Divide[1,2*Sqrt[1 + (z)^(2)]]*((Sqrt[1 + (z)^(2)]+ z)^(2*a - 1)+(Sqrt[1 + (z)^(2)]- z)^(2*a - 1))	Successful	Failure	-	Successful
15.4.E14	${\displaystyle{\displaystyle F\left(a,1-a;\tfrac{1}{2};{\sin^{2}}z\right)=\frac{\cos\left((2a-1)z\right)}{\cos z}}}$	hypergeom([a, 1 - a], [(1)/(2)], (sin(z))^(2))=(cos((2*a - 1)* z))/(cos(z))	Hypergeometric2F1[a, 1 - a, Divide[1,2], (Sin[z])^(2)]=Divide[Cos[(2*a - 1)* z],Cos[z]]	Failure	Failure	Successful	Successful
15.4.E15	${\displaystyle{\displaystyle F\left(a,1-a;\tfrac{3}{2};-z^{2}\right)=\frac{1}{(2-4a)z}\left(\left(\sqrt{1+z^{2}}+z\right)^{1-2a}-\left(\sqrt{1+z^{2}}-z\right)^{1-2a}\right)}}$	hypergeom([a, 1 - a], [(3)/(2)], - (z)^(2))=(1)/((2 - 4*a)* z)*((sqrt(1 + (z)^(2))+ z)^(1 - 2*a)-(sqrt(1 + (z)^(2))- z)^(1 - 2*a))	Hypergeometric2F1[a, 1 - a, Divide[3,2], - (z)^(2)]=Divide[1,(2 - 4*a)* z]*((Sqrt[1 + (z)^(2)]+ z)^(1 - 2*a)-(Sqrt[1 + (z)^(2)]- z)^(1 - 2*a))	Failure	Failure	Successful	Successful
15.4.E16	${\displaystyle{\displaystyle F\left(a,1-a;\tfrac{3}{2};{\sin^{2}}z\right)=\frac{\sin\left((2a-1)z\right)}{(2a-1)\sin z}}}$	hypergeom([a, 1 - a], [(3)/(2)], (sin(z))^(2))=(sin((2*a - 1)* z))/((2*a - 1)* sin(z))	Hypergeometric2F1[a, 1 - a, Divide[3,2], (Sin[z])^(2)]=Divide[Sin[(2*a - 1)* z],(2*a - 1)* Sin[z]]	Successful	Failure	-	Successful
15.4.E17	${\displaystyle{\displaystyle F\left(a,\tfrac{1}{2}+a;1+2a;z\right)=\left(\tfrac{1}{2}+\tfrac{1}{2}\sqrt{1-z}\right)^{-2a}}}$	hypergeom([a, (1)/(2)+ a], [1 + 2*a], z)=((1)/(2)+(1)/(2)*sqrt(1 - z))^(- 2*a)	Hypergeometric2F1[a, Divide[1,2]+ a, 1 + 2*a, z]=(Divide[1,2]+Divide[1,2]*Sqrt[1 - z])^(- 2*a)	Failure	Successful	Successful	-
15.4.E18	${\displaystyle{\displaystyle F\left(a,\tfrac{1}{2}+a;2a;z\right)=\frac{1}{\sqrt{1-z}}\left(\tfrac{1}{2}+\tfrac{1}{2}\sqrt{1-z}\right)^{1-2a}}}$	hypergeom([a, (1)/(2)+ a], [2*a], z)=(1)/(sqrt(1 - z))*((1)/(2)+(1)/(2)*sqrt(1 - z))^(1 - 2*a)	Hypergeometric2F1[a, Divide[1,2]+ a, 2*a, z]=Divide[1,Sqrt[1 - z]]*(Divide[1,2]+Divide[1,2]*Sqrt[1 - z])^(1 - 2*a)	Failure	Successful	Successful	-
15.4.E19	${\displaystyle{\displaystyle F\left(a+1,b;a;z\right)=\left(1-(1-(\ifrac{b}{a}))z\right)(1-z)^{-1-b}}}$	hypergeom([a + 1, b], [a], z)=(1 -(1 -((b)/(a)))*z)*(1 - z)^(- 1 - b)	Hypergeometric2F1[a + 1, b, a, z]=(1 -(1 -(Divide[b,a]))*z)*(1 - z)^(- 1 - b)	Successful	Successful	-	-
15.4.E20	${\displaystyle{\displaystyle F\left(a,b;c;1\right)=\frac{\Gamma\left(c\right)\Gamma\left(c-a-b\right)}{\Gamma\left(c-a\right)\Gamma\left(c-b\right)}}}$	hypergeom([a, b], [c], 1)=(GAMMA(c)*GAMMA(c - a - b))/(GAMMA(c - a)*GAMMA(c - b))	Hypergeometric2F1[a, b, c, 1]=Divide[Gamma[c]*Gamma[c - a - b],Gamma[c - a]*Gamma[c - b]]	Successful	Failure	-	Fail DirectedInfinity[] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
15.4.E21	${\displaystyle{\displaystyle\lim_{z\to 1-}\frac{F\left(a,b;a+b;z\right)}{-\ln\left(1-z\right)}=\frac{\Gamma\left(a+b\right)}{\Gamma\left(a\right)\Gamma\left(b\right)}}}$	limit((hypergeom([a, b], [a + b], z))/(- ln(1 - z)), z = 1, left)=(GAMMA(a + b))/(GAMMA(a)*GAMMA(b))	Limit[Divide[Hypergeometric2F1[a, b, a + b, z],- Log[1 - z]], z -> 1, Direction -> "FromBelow"]=Divide[Gamma[a + b],Gamma[a]*Gamma[b]]	Successful	Successful	-	-
15.4.E22	${\displaystyle{\displaystyle\lim_{z\to 1-}(1-z)^{a+b-c}\left(F\left(a,b;c;z\right)-\frac{\Gamma\left(c\right)\Gamma\left(c-a-b\right)}{\Gamma\left(c-a\right)\Gamma\left(c-b\right)}\right)=\frac{\Gamma\left(c\right)\Gamma\left(a+b-c\right)}{\Gamma\left(a\right)\Gamma\left(b\right)}}}$	limit((1 - z)^(a + b - c)*(hypergeom([a, b], [c], z)-(GAMMA(c)*GAMMA(c - a - b))/(GAMMA(c - a)*GAMMA(c - b))), z = 1, left)=(GAMMA(c)*GAMMA(a + b - c))/(GAMMA(a)*GAMMA(b))	Limit[(1 - z)^(a + b - c)*(Hypergeometric2F1[a, b, c, z]-Divide[Gamma[c]*Gamma[c - a - b],Gamma[c - a]*Gamma[c - b]]), z -> 1, Direction -> "FromBelow"]=Divide[Gamma[c]*Gamma[a + b - c],Gamma[a]*Gamma[b]]	Failure	Failure	Skip	Skip
15.4.E23	${\displaystyle{\displaystyle\lim_{z\to 1-}\frac{F\left(a,b;c;z\right)}{(1-z)^{c-a-b}}=\frac{\Gamma\left(c\right)\Gamma\left(a+b-c\right)}{\Gamma\left(a\right)\Gamma\left(b\right)}}}$	limit((hypergeom([a, b], [c], z))/((1 - z)^(c - a - b)), z = 1, left)=(GAMMA(c)*GAMMA(a + b - c))/(GAMMA(a)*GAMMA(b))	Limit[Divide[Hypergeometric2F1[a, b, c, z],(1 - z)^(c - a - b)], z -> 1, Direction -> "FromBelow"]=Divide[Gamma[c]*Gamma[a + b - c],Gamma[a]*Gamma[b]]	Failure	Failure	Skip	Fail DirectedInfinity[] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[a, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[a, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
15.4.E24	${\displaystyle{\displaystyle F\left(-n,b;c;1\right)=\frac{{\left(c-b\right)_{n}}}{{\left(c\right)_{n}}}}}$	hypergeom([- n, b], [c], 1)=(pochhammer(c - b, n))/(pochhammer(c, n))	Hypergeometric2F1[- n, b, c, 1]=Divide[Pochhammer[c - b, n],Pochhammer[c, n]]	Successful	Failure	-	Successful
15.4.E25	${\displaystyle{\displaystyle\sum_{n=-\infty}^{\infty}\frac{\Gamma\left(a+n\right)\Gamma\left(b+n\right)}{\Gamma\left(c+n\right)\Gamma\left(d+n\right)}=\frac{\pi^{2}}{\sin\left(\pi a\right)\sin\left(\pi b\right)}\*\frac{\Gamma\left(c+d-a-b-1\right)}{\Gamma\left(c-a\right)\Gamma\left(d-a\right)\Gamma\left(c-b\right)\Gamma\left(d-b\right)}}}$	sum((GAMMA(a + n)*GAMMA(b + n))/(GAMMA(c + n)*GAMMA(d + n)), n = - infinity..infinity)=((Pi)^(2))/(sin(Pi*a)*sin(Pi*b))*(GAMMA(c + d - a - b - 1))/(GAMMA(c - a)*GAMMA(d - a)*GAMMA(c - b)*GAMMA(d - b))	Sum[Divide[Gamma[a + n]*Gamma[b + n],Gamma[c + n]*Gamma[d + n]], {n, - Infinity, Infinity}]=Divide[(Pi)^(2),Sin[Pi*a]*Sin[Pi*b]]*Divide[Gamma[c + d - a - b - 1],Gamma[c - a]*Gamma[d - a]*Gamma[c - b]*Gamma[d - b]]	Failure	Failure	Skip	Error
15.4.E26	${\displaystyle{\displaystyle F\left(a,b;a-b+1;-1\right)=\frac{\Gamma\left(a-b+1\right)\Gamma\left(\tfrac{1}{2}a+1\right)}{\Gamma\left(a+1\right)\Gamma\left(\tfrac{1}{2}a-b+1\right)}}}$	hypergeom([a, b], [a - b + 1], - 1)=(GAMMA(a - b + 1)*GAMMA((1)/(2)*a + 1))/(GAMMA(a + 1)*GAMMA((1)/(2)*a - b + 1))	Hypergeometric2F1[a, b, a - b + 1, - 1]=Divide[Gamma[a - b + 1]*Gamma[Divide[1,2]*a + 1],Gamma[a + 1]*Gamma[Divide[1,2]*a - b + 1]]	Successful	Successful	-	-
15.4.E27	${\displaystyle{\displaystyle F\left(1,a;a+1;-1\right)=\tfrac{1}{2}a\left(\psi\left(\tfrac{1}{2}a+\tfrac{1}{2}\right)-\psi\left(\tfrac{1}{2}a\right)\right)}}$	hypergeom([1, a], [a + 1], - 1)=(1)/(2)*a*(Psi((1)/(2)*a +(1)/(2))- Psi((1)/(2)*a))	Hypergeometric2F1[1, a, a + 1, - 1]=Divide[1,2]*a*(PolyGamma[Divide[1,2]*a +Divide[1,2]]- PolyGamma[Divide[1,2]*a])	Successful	Successful	-	-
15.4.E28	${\displaystyle{\displaystyle F\left(a,b;\tfrac{1}{2}a+\tfrac{1}{2}b+\tfrac{1}{2};\tfrac{1}{2}\right)=\sqrt{\pi}\frac{\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{2}b+\tfrac{1}{2}\right)}{\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{2}\right)\Gamma\left(\tfrac{1}{2}b+\tfrac{1}{2}\right)}}}$	hypergeom([a, b], [(1)/(2)*a +(1)/(2)*b +(1)/(2)], (1)/(2))=sqrt(Pi)*(GAMMA((1)/(2)*a +(1)/(2)*b +(1)/(2)))/(GAMMA((1)/(2)*a +(1)/(2))*GAMMA((1)/(2)*b +(1)/(2)))	Hypergeometric2F1[a, b, Divide[1,2]*a +Divide[1,2]*b +Divide[1,2], Divide[1,2]]=Sqrt[Pi]*Divide[Gamma[Divide[1,2]*a +Divide[1,2]*b +Divide[1,2]],Gamma[Divide[1,2]*a +Divide[1,2]]*Gamma[Divide[1,2]*b +Divide[1,2]]]	Successful	Successful	-	-
15.4.E29	${\displaystyle{\displaystyle F\left(a,b;\tfrac{1}{2}a+\tfrac{1}{2}b+1;\tfrac{1}{2}\right)=\frac{2\sqrt{\pi}}{a-b}\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{2}b+1\right)\*\left(\frac{1}{\Gamma\left(\tfrac{1}{2}a\right)\Gamma\left(\tfrac{1}{2}b+\tfrac{1}{2}\right)}-\frac{1}{\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{2}\right)\Gamma\left(\tfrac{1}{2}b\right)}\right)}}$	hypergeom([a, b], [(1)/(2)*a +(1)/(2)*b + 1], (1)/(2))=(2*sqrt(Pi))/(a - b)*GAMMA((1)/(2)*a +(1)/(2)*b + 1)*((1)/(GAMMA((1)/(2)*a)*GAMMA((1)/(2)*b +(1)/(2)))-(1)/(GAMMA((1)/(2)*a +(1)/(2))*GAMMA((1)/(2)*b)))	Hypergeometric2F1[a, b, Divide[1,2]*a +Divide[1,2]*b + 1, Divide[1,2]]=Divide[2*Sqrt[Pi],a - b]*Gamma[Divide[1,2]*a +Divide[1,2]*b + 1]*(Divide[1,Gamma[Divide[1,2]*a]*Gamma[Divide[1,2]*b +Divide[1,2]]]-Divide[1,Gamma[Divide[1,2]*a +Divide[1,2]]*Gamma[Divide[1,2]*b]])	Failure	Failure	Fail Float(undefined)+Float(undefined)*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2)} Float(undefined)+Float(undefined)*I <- {a = 2^(1/2)-I*2^(1/2), b = 2^(1/2)-I*2^(1/2)} Float(undefined)+Float(undefined)*I <- {a = -2^(1/2)-I*2^(1/2), b = -2^(1/2)-I*2^(1/2)} Float(undefined)+Float(undefined)*I <- {a = -2^(1/2)+I*2^(1/2), b = -2^(1/2)+I*2^(1/2)}	Successful
15.4.E30	${\displaystyle{\displaystyle F\left(a,1-a;b;\tfrac{1}{2}\right)=\frac{2^{1-b}\sqrt{\pi}\Gamma\left(b\right)}{\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{2}b\right)\Gamma\left(\tfrac{1}{2}b-\tfrac{1}{2}a+\tfrac{1}{2}\right)}}}$	hypergeom([a, 1 - a], [b], (1)/(2))=((2)^(1 - b)*sqrt(Pi)*GAMMA(b))/(GAMMA((1)/(2)*a +(1)/(2)*b)*GAMMA((1)/(2)*b -(1)/(2)*a +(1)/(2)))	Hypergeometric2F1[a, 1 - a, b, Divide[1,2]]=Divide[(2)^(1 - b)*Sqrt[Pi]*Gamma[b],Gamma[Divide[1,2]*a +Divide[1,2]*b]*Gamma[Divide[1,2]*b -Divide[1,2]*a +Divide[1,2]]]	Successful	Failure	-	Successful
15.4.E31	${\displaystyle{\displaystyle F\left(a,\tfrac{1}{2}+a;\tfrac{3}{2}-2a;-\tfrac{1}{3}\right)=\left(\frac{8}{9}\right)^{-2a}\frac{\Gamma\left(\tfrac{4}{3}\right)\Gamma\left(\tfrac{3}{2}-2a\right)}{\Gamma\left(\tfrac{3}{2}\right)\Gamma\left(\tfrac{4}{3}-2a\right)}}}$	hypergeom([a, (1)/(2)+ a], [(3)/(2)- 2*a], -(1)/(3))=((8)/(9))^(- 2*a)*(GAMMA((4)/(3))*GAMMA((3)/(2)- 2*a))/(GAMMA((3)/(2))*GAMMA((4)/(3)- 2*a))	Hypergeometric2F1[a, Divide[1,2]+ a, Divide[3,2]- 2*a, -Divide[1,3]]=(Divide[8,9])^(- 2*a)*Divide[Gamma[Divide[4,3]]*Gamma[Divide[3,2]- 2*a],Gamma[Divide[3,2]]*Gamma[Divide[4,3]- 2*a]]	Failure	Failure	Successful	Successful
15.4.E32	${\displaystyle{\displaystyle F\left(a,\tfrac{1}{2}+a;\tfrac{5}{6}+\tfrac{2}{3}a;\tfrac{1}{9}\right)=\sqrt{\pi}\left(\frac{3}{4}\right)^{a}\frac{\Gamma\left(\tfrac{5}{6}+\tfrac{2}{3}a\right)}{\Gamma\left(\tfrac{1}{2}+\tfrac{1}{3}a\right)\Gamma\left(\tfrac{5}{6}+\tfrac{1}{3}a\right)}}}$	hypergeom([a, (1)/(2)+ a], [(5)/(6)+(2)/(3)*a], (1)/(9))=sqrt(Pi)*((3)/(4))^(a)*(GAMMA((5)/(6)+(2)/(3)*a))/(GAMMA((1)/(2)+(1)/(3)*a)*GAMMA((5)/(6)+(1)/(3)*a))	Hypergeometric2F1[a, Divide[1,2]+ a, Divide[5,6]+Divide[2,3]*a, Divide[1,9]]=Sqrt[Pi]*(Divide[3,4])^(a)*Divide[Gamma[Divide[5,6]+Divide[2,3]*a],Gamma[Divide[1,2]+Divide[1,3]*a]*Gamma[Divide[5,6]+Divide[1,3]*a]]	Failure	Failure	Successful	Successful
15.4.E33	${\displaystyle{\displaystyle F\left(3a,\tfrac{1}{3}+a;\tfrac{2}{3}+2a;e^{\ifrac{\mathrm{i}\pi}{3}}\right)=\sqrt{\pi}e^{\ifrac{\mathrm{i}\pi a}{2}}\left(\frac{16}{27}\right)^{(3a+1)/6}\frac{\Gamma\left(\frac{5}{6}+a\right)}{\Gamma\left(\frac{2}{3}+a\right)\Gamma\left(\frac{2}{3}\right)}}}$	hypergeom([3*a, (1)/(3)+ a], [(2)/(3)+ 2*a], exp((I*Pi)/(3)))=sqrt(Pi)*exp((I*Pi*a)/(2))*((16)/(27))^((3*a + 1)/ 6)*(GAMMA((5)/(6)+ a))/(GAMMA((2)/(3)+ a)*GAMMA((2)/(3)))	Hypergeometric2F1[3*a, Divide[1,3]+ a, Divide[2,3]+ 2*a, Exp[Divide[I*Pi,3]]]=Sqrt[Pi]*Exp[Divide[I*Pi*a,2]]*(Divide[16,27])^((3*a + 1)/ 6)*Divide[Gamma[Divide[5,6]+ a],Gamma[Divide[2,3]+ a]*Gamma[Divide[2,3]]]	Failure	Failure	Successful	Successful
15.5.E1	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}F\left(a,b;c;z\right)=\frac{ab}{c}F\left(a+1,b+1;c+1;z\right)}}$	diff(hypergeom([a, b], [c], z), z)=(a*b)/(c)*hypergeom([a + 1, b + 1], [c + 1], z)	D[Hypergeometric2F1[a, b, c, z], z]=Divide[a*b,c]*Hypergeometric2F1[a + 1, b + 1, c + 1, z]	Successful	Successful	-	-
15.5.E2	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}F\left(a,b;c;z\right)=\frac{{\left(a\right)_{n}}{\left(b\right)_{n}}}{{\left(c\right)_{n}}}\*F\left(a+n,b+n;c+n;z\right)}}$	diff(hypergeom([a, b], [c], z), [z$(n)])=(pochhammer(a, n)*pochhammer(b, n))/(pochhammer(c, n))* hypergeom([a + n, b + n], [c + n], z)	D[Hypergeometric2F1[a, b, c, z], {z, n}]=Divide[Pochhammer[a, n]*Pochhammer[b, n],Pochhammer[c, n]]* Hypergeometric2F1[a + n, b + n, c + n, z]	Successful	Successful	-	-
15.5.E3	${\displaystyle{\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(z^{a-1}F\left(a,b;c;z\right)\right)={\left(a\right)_{n}}z^{a+n-1}F\left(a+n,b;c;z\right)}}$	(z*diff(z, z))^(n)*((z)^(a - 1)* hypergeom([a, b], [c], z))= pochhammer(a, n)*(z)^(a + n - 1)* hypergeom([a + n, b], [c], z)	(z*D[z, z])^(n)*((z)^(a - 1)* Hypergeometric2F1[a, b, c, z])= Pochhammer[a, n]*(z)^(a + n - 1)* Hypergeometric2F1[a + n, b, c, z]	Failure	Failure	Fail -.6225095031e-1-.5644186814e-1*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 1} .1785845766+.2832629970e-1*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 2} -.7314115943-.4867699196*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 3} -34.57094218-72.21034026*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), n = 1} ... skip entries to safe data	Skip
15.5.E4	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^{c-1}F\left(a,b;c;z\right)\right)={\left(c-n\right)_{n}}z^{c-n-1}F\left(a,b;c-n;z\right)}}$	diff((z)^(c - 1)* hypergeom([a, b], [c], z), [z$(n)])= pochhammer(c - n, n)*(z)^(c - n - 1)* hypergeom([a, b], [c - n], z)	D[(z)^(c - 1)* Hypergeometric2F1[a, b, c, z], {z, n}]= Pochhammer[c - n, n]*(z)^(c - n - 1)* Hypergeometric2F1[a, b, c - n, z]	Failure	Failure	Skip	Skip
15.5.E5	${\displaystyle{\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(z^{c-a-1}(1-z)^{a+b-c}F\left(a,b;c;z\right)\right)={\left(c-a\right)_{n}}z^{c-a+n-1}(1-z)^{a-n+b-c}\*F\left(a-n,b;c;z\right)}}$	(z*diff(z, z))^(n)*((z)^(c - a - 1)*(1 - z)^(a + b - c)* hypergeom([a, b], [c], z))= pochhammer(c - a, n)*(z)^(c - a + n - 1)*(1 - z)^(a - n + b - c)* hypergeom([a - n, b], [c], z)	(z*D[z, z])^(n)*((z)^(c - a - 1)*(1 - z)^(a + b - c)* Hypergeometric2F1[a, b, c, z])= Pochhammer[c - a, n]*(z)^(c - a + n - 1)*(1 - z)^(a - n + b - c)* Hypergeometric2F1[a - n, b, c, z]	Failure	Failure	Fail 1.000000000-.2828427124e-9*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 1} 1.414213562+1.414213562*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 2} .1131370849e-8+3.999999998*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 3} 1.000000000+0.*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), n = 1} ... skip entries to safe data	Skip
15.5.E6	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left((1-z)^{a+b-c}F\left(a,b;c;z\right)\right)=\frac{{\left(c-a\right)_{n}}{\left(c-b\right)_{n}}}{{\left(c\right)_{n}}}(1-z)^{a+b-c-n}\*F\left(a,b;c+n;z\right)}}$	diff((1 - z)^(a + b - c)* hypergeom([a, b], [c], z), [z$(n)])=(pochhammer(c - a, n)*pochhammer(c - b, n))/(pochhammer(c, n))*(1 - z)^(a + b - c - n)* hypergeom([a, b], [c + n], z)	D[(1 - z)^(a + b - c)* Hypergeometric2F1[a, b, c, z], {z, n}]=Divide[Pochhammer[c - a, n]*Pochhammer[c - b, n],Pochhammer[c, n]]*(1 - z)^(a + b - c - n)* Hypergeometric2F1[a, b, c + n, z]	Failure	Failure	Skip	Skip
15.5.E7	${\displaystyle{\displaystyle\left((1-z)\frac{\mathrm{d}}{\mathrm{d}z}(1-z)\right)^{n}\left((1-z)^{a-1}F\left(a,b;c;z\right)\right)=(-1)^{n}\frac{{\left(a\right)_{n}}{\left(c-b\right)_{n}}}{{\left(c\right)_{n}}}(1-z)^{a+n-1}\*F\left(a+n,b;c+n;z\right)}}$	((1 - z)*diff(1 - z, z))^(n)*((1 - z)^(a - 1)* hypergeom([a, b], [c], z))=(- 1)^(n)*(pochhammer(a, n)*pochhammer(c - b, n))/(pochhammer(c, n))*(1 - z)^(a + n - 1)* hypergeom([a + n, b], [c + n], z)	((1 - z)*D[1 - z, z])^(n)*((1 - z)^(a - 1)* Hypergeometric2F1[a, b, c, z])=(- 1)^(n)*Divide[Pochhammer[a, n]*Pochhammer[c - b, n],Pochhammer[c, n]]*(1 - z)^(a + n - 1)* Hypergeometric2F1[a + n, b, c + n, z]	Failure	Failure	Fail -1.000000000+.574091994e-10*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 1} -.4142135623-1.414213562*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 2} 1.828427125-1.171572874*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 3} -1.000000000+.254335130e-10*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), n = 1} ... skip entries to safe data	Skip
15.5.E8	${\displaystyle{\displaystyle\left((1-z)\frac{\mathrm{d}}{\mathrm{d}z}(1-z)\right)^{n}\left(z^{c-1}(1-z)^{b-c}F\left(a,b;c;z\right)\right)={\left(c-n\right)_{n}}z^{c-n-1}(1-z)^{b-c+n}\*F\left(a-n,b;c-n;z\right)}}$	((1 - z)*diff(1 - z, z))^(n)*((z)^(c - 1)*(1 - z)^(b - c)* hypergeom([a, b], [c], z))= pochhammer(c - n, n)*(z)^(c - n - 1)*(1 - z)^(b - c + n)* hypergeom([a - n, b], [c - n], z)	((1 - z)*D[1 - z, z])^(n)*((z)^(c - 1)*(1 - z)^(b - c)* Hypergeometric2F1[a, b, c, z])= Pochhammer[c - n, n]*(z)^(c - n - 1)*(1 - z)^(b - c + n)* Hypergeometric2F1[a - n, b, c - n, z]	Failure	Failure	Fail .7567079467e-2-.4498677196e-1*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 1} .3281146176e-1-.3488816280e-1*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 2} .1860334783e-1+.1631943039e-1*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 3} -43.13343073-5.34730687*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), n = 1} ... skip entries to safe data	Skip
15.5.E9	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^{c-1}(1-z)^{a+b-c}F\left(a,b;c;z\right)\right)={\left(c-n\right)_{n}}z^{c-n-1}(1-z)^{a+b-c-n}\*F\left(a-n,b-n;c-n;z\right)}}$	diff((z)^(c - 1)*(1 - z)^(a + b - c)* hypergeom([a, b], [c], z), [z$(n)])= pochhammer(c - n, n)*(z)^(c - n - 1)*(1 - z)^(a + b - c - n)* hypergeom([a - n, b - n], [c - n], z)	D[(z)^(c - 1)*(1 - z)^(a + b - c)* Hypergeometric2F1[a, b, c, z], {z, n}]= Pochhammer[c - n, n]*(z)^(c - n - 1)*(1 - z)^(a + b - c - n)* Hypergeometric2F1[a - n, b - n, c - n, z]	Failure	Failure	Skip	Skip
15.5.E10	${\displaystyle{\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}=z^{n}\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}z^{n}}}$	(z*diff(z, z))^(n)= (z)^(n)* diff((z)^(n), [z$(n)])	(z*D[z, z])^(n)= (z)^(n)* D[(z)^(n), {z, n}]	Failure	Failure	Fail 28.28427122-28.28427122*I <- {z = 2^(1/2)+I*2^(1/2), n = 3} 28.28427122+28.28427122*I <- {z = 2^(1/2)-I*2^(1/2), n = 3} -28.28427122+28.28427122*I <- {z = -2^(1/2)-I*2^(1/2), n = 3} -28.28427122-28.28427122*I <- {z = -2^(1/2)+I*2^(1/2), n = 3}	Fail Complex[28.284271247461902, -28.284271247461902] <- {Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[28.284271247461902, 28.284271247461902] <- {Rule[n, 3], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-28.284271247461902, 28.284271247461902] <- {Rule[n, 3], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-28.284271247461902, -28.284271247461902] <- {Rule[n, 3], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
15.5.E11	${\displaystyle{\displaystyle(c-a)F\left(a-1,b;c;z\right)+\left(2a-c+(b-a)z\right)F\left(a,b;c;z\right)+a(z-1)F\left(a+1,b;c;z\right)=0}}$	(c - a)* hypergeom([a - 1, b], [c], z)+(2*a - c +(b - a)*z)* hypergeom([a, b], [c], z)+ a*(z - 1)* hypergeom([a + 1, b], [c], z)= 0	(c - a)* Hypergeometric2F1[a - 1, b, c, z]+(2*a - c +(b - a)*z)* Hypergeometric2F1[a, b, c, z]+ a*(z - 1)* Hypergeometric2F1[a + 1, b, c, z]= 0	Successful	Successful	-	-
15.5.E12	${\displaystyle{\displaystyle(b-a)F\left(a,b;c;z\right)+aF\left(a+1,b;c;z\right)-bF\left(a,b+1;c;z\right)=0}}$	(b - a)* hypergeom([a, b], [c], z)+ a*hypergeom([a + 1, b], [c], z)- b*hypergeom([a, b + 1], [c], z)= 0	(b - a)* Hypergeometric2F1[a, b, c, z]+ a*Hypergeometric2F1[a + 1, b, c, z]- b*Hypergeometric2F1[a, b + 1, c, z]= 0	Successful	Successful	-	-
15.5.E13	${\displaystyle{\displaystyle(c-a-b)F\left(a,b;c;z\right)+a(1-z)F\left(a+1,b;c;z\right)-(c-b)F\left(a,b-1;c;z\right)=0}}$	(c - a - b)* hypergeom([a, b], [c], z)+ a*(1 - z)* hypergeom([a + 1, b], [c], z)-(c - b)* hypergeom([a, b - 1], [c], z)= 0	(c - a - b)* Hypergeometric2F1[a, b, c, z]+ a*(1 - z)* Hypergeometric2F1[a + 1, b, c, z]-(c - b)* Hypergeometric2F1[a, b - 1, c, z]= 0	Successful	Successful	-	-
15.5.E14	${\displaystyle{\displaystyle c\left(a+(b-c)z\right)F\left(a,b;c;z\right)-ac(1-z)F\left(a+1,b;c;z\right)+(c-a)(c-b)zF\left(a,b;c+1;z\right)=0}}$	c*(a +(b - c)*z)* hypergeom([a, b], [c], z)- a*c*(1 - z)* hypergeom([a + 1, b], [c], z)+(c - a)*(c - b)* z*hypergeom([a, b], [c + 1], z)= 0	c*(a +(b - c)*z)* Hypergeometric2F1[a, b, c, z]- a*c*(1 - z)* Hypergeometric2F1[a + 1, b, c, z]+(c - a)*(c - b)* z*Hypergeometric2F1[a, b, c + 1, z]= 0	Successful	Successful	-	-
15.5.E15	${\displaystyle{\displaystyle(c-a-1)F\left(a,b;c;z\right)+aF\left(a+1,b;c;z\right)-(c-1)F\left(a,b;c-1;z\right)=0}}$	(c - a - 1)* hypergeom([a, b], [c], z)+ a*hypergeom([a + 1, b], [c], z)-(c - 1)* hypergeom([a, b], [c - 1], z)= 0	(c - a - 1)* Hypergeometric2F1[a, b, c, z]+ a*Hypergeometric2F1[a + 1, b, c, z]-(c - 1)* Hypergeometric2F1[a, b, c - 1, z]= 0	Successful	Successful	-	-
15.5.E16	${\displaystyle{\displaystyle c(1-z)F\left(a,b;c;z\right)-cF\left(a-1,b;c;z\right)+(c-b)zF\left(a,b;c+1;z\right)=0}}$	c*(1 - z)* hypergeom([a, b], [c], z)- c*hypergeom([a - 1, b], [c], z)+(c - b)* z*hypergeom([a, b], [c + 1], z)= 0	c*(1 - z)* Hypergeometric2F1[a, b, c, z]- c*Hypergeometric2F1[a - 1, b, c, z]+(c - b)* z*Hypergeometric2F1[a, b, c + 1, z]= 0	Successful	Successful	-	-
15.5.E17	${\displaystyle{\displaystyle\left(a-1+(b+1-c)z\right)F\left(a,b;c;z\right)+(c-a)F\left(a-1,b;c;z\right)-(c-1)(1-z)F\left(a,b;c-1;z\right)=0}}$	(a - 1 +(b + 1 - c)*z)* hypergeom([a, b], [c], z)+(c - a)* hypergeom([a - 1, b], [c], z)-(c - 1)*(1 - z)* hypergeom([a, b], [c - 1], z)= 0	(a - 1 +(b + 1 - c)*z)* Hypergeometric2F1[a, b, c, z]+(c - a)* Hypergeometric2F1[a - 1, b, c, z]-(c - 1)*(1 - z)* Hypergeometric2F1[a, b, c - 1, z]= 0	Successful	Successful	-	-
15.5.E18	${\displaystyle{\displaystyle c(c-1)(z-1)F\left(a,b;c-1;z\right)+{c\left(c-1-(2c-a-b-1)z\right)}F\left(a,b;c;z\right)+(c-a)(c-b)zF\left(a,b;c+1;z\right)=0}}$	c*(c - 1)*(z - 1)* hypergeom([a, b], [c - 1], z)+c*(c - 1 -(2*c - a - b - 1)*z)*hypergeom([a, b], [c], z)+(c - a)*(c - b)* z*hypergeom([a, b], [c + 1], z)= 0	c*(c - 1)*(z - 1)* Hypergeometric2F1[a, b, c - 1, z]+c*(c - 1 -(2*c - a - b - 1)*z)*Hypergeometric2F1[a, b, c, z]+(c - a)*(c - b)* z*Hypergeometric2F1[a, b, c + 1, z]= 0	Successful	Successful	-	-
15.5.E19	${\displaystyle{\displaystyle{z(1-z)(a+1)(b+1)}F\left(a+2,b+2;c+2;z\right)+{(c-(a+b+1)z)(c+1)}F\left(a+1,b+1;c+1;z\right)-{c(c+1)}F\left(a,b;c;z\right)=0}}$	z*(1 - z)*(a + 1)*(b + 1)*hypergeom([a + 2, b + 2], [c + 2], z)+(c -(a + b + 1)*z)*(c + 1)*hypergeom([a + 1, b + 1], [c + 1], z)-c*(c + 1)*hypergeom([a, b], [c], z)= 0	z*(1 - z)*(a + 1)*(b + 1)*Hypergeometric2F1[a + 2, b + 2, c + 2, z]+(c -(a + b + 1)*z)*(c + 1)*Hypergeometric2F1[a + 1, b + 1, c + 1, z]-c*(c + 1)*Hypergeometric2F1[a, b, c, z]= 0	Successful	Successful	-	-
15.5.E20	${\displaystyle{\displaystyle z(1-z)\left(\ifrac{\mathrm{d}F\left(a,b;c;z\right)}{\mathrm{d}z}\right)=(c-a)F\left(a-1,b;c;z\right)+(a-c+bz)F\left(a,b;c;z\right)}}$	z*(1 - z)*(diff(hypergeom([a, b], [c], z), z))=(c - a)* hypergeom([a - 1, b], [c], z)+(a - c + b*z)* hypergeom([a, b], [c], z)	z*(1 - z)*(D[Hypergeometric2F1[a, b, c, z], z])=(c - a)* Hypergeometric2F1[a - 1, b, c, z]+(a - c + b*z)* Hypergeometric2F1[a, b, c, z]	Successful	Successful	-	-
15.5.E20	${\displaystyle{\displaystyle(c-a)F\left(a-1,b;c;z\right)+(a-c+bz)F\left(a,b;c;z\right)=(c-b)F\left(a,b-1;c;z\right)+(b-c+az)F\left(a,b;c;z\right)}}$	(c - a)* hypergeom([a - 1, b], [c], z)+(a - c + b*z)* hypergeom([a, b], [c], z)=(c - b)* hypergeom([a, b - 1], [c], z)+(b - c + a*z)* hypergeom([a, b], [c], z)	(c - a)* Hypergeometric2F1[a - 1, b, c, z]+(a - c + b*z)* Hypergeometric2F1[a, b, c, z]=(c - b)* Hypergeometric2F1[a, b - 1, c, z]+(b - c + a*z)* Hypergeometric2F1[a, b, c, z]	Successful	Successful	-	-
15.5.E21	${\displaystyle{\displaystyle c(1-z)\left(\ifrac{\mathrm{d}F\left(a,b;c;z\right)}{\mathrm{d}z}\right)=(c-a)(c-b)F\left(a,b;c+1;z\right)+c(a+b-c)F\left(a,b;c;z\right)}}$	c*(1 - z)*(diff(hypergeom([a, b], [c], z), z))=(c - a)*(c - b)* hypergeom([a, b], [c + 1], z)+ c*(a + b - c)* hypergeom([a, b], [c], z)	c*(1 - z)*(D[Hypergeometric2F1[a, b, c, z], z])=(c - a)*(c - b)* Hypergeometric2F1[a, b, c + 1, z]+ c*(a + b - c)* Hypergeometric2F1[a, b, c, z]	Successful	Successful	-	-
15.8.E1	${\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop c};z\right)=(1-z)^{-a}\mathbf{F}\left({a,c-b\atop c};\frac{z}{z-1}\right)}}$	hypergeom([a, b], [c], z)/GAMMA(c)=(1 - z)^(- a)* hypergeom([a, c - b], [c], (z)/(z - 1))/GAMMA(c)	Hypergeometric2F1Regularized[a, b, c, z]=(1 - z)^(- a)* Hypergeometric2F1Regularized[a, c - b, c, Divide[z,z - 1]]	Failure	Failure	Skip	Skip
15.8.E1	${\displaystyle{\displaystyle(1-z)^{-a}\mathbf{F}\left({a,c-b\atop c};\frac{z}{z-1}\right)=(1-z)^{-b}\mathbf{F}\left({c-a,b\atop c};\frac{z}{z-1}\right)}}$	(1 - z)^(- a)* hypergeom([a, c - b], [c], (z)/(z - 1))/GAMMA(c)=(1 - z)^(- b)* hypergeom([c - a, b], [c], (z)/(z - 1))/GAMMA(c)	(1 - z)^(- a)* Hypergeometric2F1Regularized[a, c - b, c, Divide[z,z - 1]]=(1 - z)^(- b)* Hypergeometric2F1Regularized[c - a, b, c, Divide[z,z - 1]]	Failure	Failure	Skip	Skip
15.8.E1	${\displaystyle{\displaystyle(1-z)^{-b}\mathbf{F}\left({c-a,b\atop c};\frac{z}{z-1}\right)=(1-z)^{c-a-b}\mathbf{F}\left({c-a,c-b\atop c};z\right)}}$	(1 - z)^(- b)* hypergeom([c - a, b], [c], (z)/(z - 1))/GAMMA(c)=(1 - z)^(c - a - b)* hypergeom([c - a, c - b], [c], z)/GAMMA(c)	(1 - z)^(- b)* Hypergeometric2F1Regularized[c - a, b, c, Divide[z,z - 1]]=(1 - z)^(c - a - b)* Hypergeometric2F1Regularized[c - a, c - b, c, z]	Failure	Failure	Skip	Skip
15.8.E2	${\displaystyle{\displaystyle\frac{\sin\left(\pi(b-a)\right)}{\pi}\mathbf{F}\left({a,b\atop c};z\right)=\frac{(-z)^{-a}}{\Gamma\left(b\right)\Gamma\left(c-a\right)}\mathbf{F}\left({a,a-c+1\atop a-b+1};\frac{1}{z}\right)-\frac{(-z)^{-b}}{\Gamma\left(a\right)\Gamma\left(c-b\right)}\mathbf{F}\left({b,b-c+1\atop b-a+1};\frac{1}{z}\right)}}$	(sin(Pi*(b - a)))/(Pi)*hypergeom([a, b], [c], z)/GAMMA(c)=((- z)^(- a))/(GAMMA(b)*GAMMA(c - a))*hypergeom([a, a - c + 1], [a - b + 1], (1)/(z))/GAMMA(a - b + 1)-((- z)^(- b))/(GAMMA(a)*GAMMA(c - b))*hypergeom([b, b - c + 1], [b - a + 1], (1)/(z))/GAMMA(b - a + 1)	Divide[Sin[Pi*(b - a)],Pi]*Hypergeometric2F1Regularized[a, b, c, z]=Divide[(- z)^(- a),Gamma[b]*Gamma[c - a]]*Hypergeometric2F1Regularized[a, a - c + 1, a - b + 1, Divide[1,z]]-Divide[(- z)^(- b),Gamma[a]*Gamma[c - b]]*Hypergeometric2F1Regularized[b, b - c + 1, b - a + 1, Divide[1,z]]	Failure	Failure	Skip	Skip
15.8.E3	${\displaystyle{\displaystyle\frac{\sin\left(\pi(b-a)\right)}{\pi}\mathbf{F}\left({a,b\atop c};z\right)=\frac{(1-z)^{-a}}{\Gamma\left(b\right)\Gamma\left(c-a\right)}\mathbf{F}\left({a,c-b\atop a-b+1};\frac{1}{1-z}\right)-\frac{(1-z)^{-b}}{\Gamma\left(a\right)\Gamma\left(c-b\right)}\mathbf{F}\left({b,c-a\atop b-a+1};\frac{1}{1-z}\right)}}$	(sin(Pi*(b - a)))/(Pi)*hypergeom([a, b], [c], z)/GAMMA(c)=((1 - z)^(- a))/(GAMMA(b)*GAMMA(c - a))*hypergeom([a, c - b], [a - b + 1], (1)/(1 - z))/GAMMA(a - b + 1)-((1 - z)^(- b))/(GAMMA(a)*GAMMA(c - b))*hypergeom([b, c - a], [b - a + 1], (1)/(1 - z))/GAMMA(b - a + 1)	Divide[Sin[Pi*(b - a)],Pi]*Hypergeometric2F1Regularized[a, b, c, z]=Divide[(1 - z)^(- a),Gamma[b]*Gamma[c - a]]*Hypergeometric2F1Regularized[a, c - b, a - b + 1, Divide[1,1 - z]]-Divide[(1 - z)^(- b),Gamma[a]*Gamma[c - b]]*Hypergeometric2F1Regularized[b, c - a, b - a + 1, Divide[1,1 - z]]	Failure	Failure	Skip	Skip
15.8.E4	${\displaystyle{\displaystyle\frac{\sin\left(\pi(c-a-b)\right)}{\pi}\mathbf{F}\left({a,b\atop c};z\right)=\frac{1}{\Gamma\left(c-a\right)\Gamma\left(c-b\right)}\mathbf{F}\left({a,b\atop a+b-c+1};1-z\right)-\frac{(1-z)^{c-a-b}}{\Gamma\left(a\right)\Gamma\left(b\right)}\mathbf{F}\left({c-a,c-b\atop c-a-b+1};1-z\right)}}$	(sin(Pi*(c - a - b)))/(Pi)*hypergeom([a, b], [c], z)/GAMMA(c)=(1)/(GAMMA(c - a)*GAMMA(c - b))*hypergeom([a, b], [a + b - c + 1], 1 - z)/GAMMA(a + b - c + 1)-((1 - z)^(c - a - b))/(GAMMA(a)*GAMMA(b))*hypergeom([c - a, c - b], [c - a - b + 1], 1 - z)/GAMMA(c - a - b + 1)	Divide[Sin[Pi*(c - a - b)],Pi]*Hypergeometric2F1Regularized[a, b, c, z]=Divide[1,Gamma[c - a]*Gamma[c - b]]*Hypergeometric2F1Regularized[a, b, a + b - c + 1, 1 - z]-Divide[(1 - z)^(c - a - b),Gamma[a]*Gamma[b]]*Hypergeometric2F1Regularized[c - a, c - b, c - a - b + 1, 1 - z]	Failure	Failure	Skip	Skip
15.8.E5	${\displaystyle{\displaystyle\frac{\sin\left(\pi(c-a-b)\right)}{\pi}\mathbf{F}\left({a,b\atop c};z\right)=\frac{z^{-a}}{\Gamma\left(c-a\right)\Gamma\left(c-b\right)}\mathbf{F}\left({a,a-c+1\atop a+b-c+1};1-\frac{1}{z}\right)-\frac{(1-z)^{c-a-b}z^{a-c}}{\Gamma\left(a\right)\Gamma\left(b\right)}\mathbf{F}\left({c-a,1-a\atop c-a-b+1};1-\frac{1}{z}\right)}}$	(sin(Pi*(c - a - b)))/(Pi)*hypergeom([a, b], [c], z)/GAMMA(c)=((z)^(- a))/(GAMMA(c - a)*GAMMA(c - b))*hypergeom([a, a - c + 1], [a + b - c + 1], 1 -(1)/(z))/GAMMA(a + b - c + 1)-((1 - z)^(c - a - b)* (z)^(a - c))/(GAMMA(a)*GAMMA(b))*hypergeom([c - a, 1 - a], [c - a - b + 1], 1 -(1)/(z))/GAMMA(c - a - b + 1)	Divide[Sin[Pi*(c - a - b)],Pi]*Hypergeometric2F1Regularized[a, b, c, z]=Divide[(z)^(- a),Gamma[c - a]*Gamma[c - b]]*Hypergeometric2F1Regularized[a, a - c + 1, a + b - c + 1, 1 -Divide[1,z]]-Divide[(1 - z)^(c - a - b)* (z)^(a - c),Gamma[a]*Gamma[b]]*Hypergeometric2F1Regularized[c - a, 1 - a, c - a - b + 1, 1 -Divide[1,z]]	Failure	Failure	Skip	Skip
15.8.E6	${\displaystyle{\displaystyle F\left({-m,b\atop c};z\right)=\frac{(b)_{m}}{(c)_{m}}(-z)^{m}F\left({-m,1-c-m\atop 1-b-m};\frac{1}{z}\right)}}$	hypergeom([- m, b], [c], z)=(b[m])/(c[m])*(- z)^(m)* hypergeom([- m, 1 - c - m], [1 - b - m], (1)/(z))	Hypergeometric2F1[- m, b, c, z]=Divide[Subscript[b, m],Subscript[c, m]]*(- z)^(m)* Hypergeometric2F1[- m, 1 - c - m, 1 - b - m, Divide[1,z]]	Failure	Failure	Skip	Skip
15.8.E6	${\displaystyle{\displaystyle\frac{(b)_{m}}{(c)_{m}}(-z)^{m}F\left({-m,1-c-m\atop 1-b-m};\frac{1}{z}\right)=\frac{(b)_{m}}{(c)_{m}}(1-z)^{m}F\left({-m,c-b\atop 1-b-m};\frac{1}{1-z}\right)}}$	(b[m])/(c[m])*(- z)^(m)* hypergeom([- m, 1 - c - m], [1 - b - m], (1)/(z))=(b[m])/(c[m])*(1 - z)^(m)* hypergeom([- m, c - b], [1 - b - m], (1)/(1 - z))	Divide[Subscript[b, m],Subscript[c, m]]*(- z)^(m)* Hypergeometric2F1[- m, 1 - c - m, 1 - b - m, Divide[1,z]]=Divide[Subscript[b, m],Subscript[c, m]]*(1 - z)^(m)* Hypergeometric2F1[- m, c - b, 1 - b - m, Divide[1,1 - z]]	Failure	Failure	Skip	Skip
15.8.E7	${\displaystyle{\displaystyle F\left({-m,b\atop c};z\right)=\frac{(c-b)_{m}}{(c)_{m}}F\left({-m,b\atop b-c-m+1};1-z\right)}}$	hypergeom([- m, b], [c], z)=(c - b[m])/(c[m])*hypergeom([- m, b], [b - c - m + 1], 1 - z)	Hypergeometric2F1[- m, b, c, z]=Divide[Subscript[c - b, m],Subscript[c, m]]*Hypergeometric2F1[- m, b, b - c - m + 1, 1 - z]	Failure	Failure	Skip	Skip
15.8.E7	${\displaystyle{\displaystyle\frac{(c-b)_{m}}{(c)_{m}}F\left({-m,b\atop b-c-m+1};1-z\right)=\frac{(c-b)_{m}}{(c)_{m}}z^{m}F\left({-m,1-c-m\atop b-c-m+1};1-\frac{1}{z}\right)}}$	(c - b[m])/(c[m])*hypergeom([- m, b], [b - c - m + 1], 1 - z)=(c - b[m])/(c[m])*(z)^(m)* hypergeom([- m, 1 - c - m], [b - c - m + 1], 1 -(1)/(z))	Divide[Subscript[c - b, m],Subscript[c, m]]*Hypergeometric2F1[- m, b, b - c - m + 1, 1 - z]=Divide[Subscript[c - b, m],Subscript[c, m]]*(z)^(m)* Hypergeometric2F1[- m, 1 - c - m, b - c - m + 1, 1 -Divide[1,z]]	Failure	Failure	Skip	Skip
15.8.E13	${\displaystyle{\displaystyle F\left({a,b\atop 2b};z\right)=\left(1-\tfrac{1}{2}z\right)^{-a}F\left({\tfrac{1}{2}a,\tfrac{1}{2}a+\tfrac{1}{2}\atop b+\tfrac{1}{2}};\left(\frac{z}{2-z}\right)^{2}\right)}}$	hypergeom([a, b], [2*b], z)=(1 -(1)/(2)*z)^(- a)* hypergeom([(1)/(2)*a, (1)/(2)*a +(1)/(2)], [b +(1)/(2)], ((z)/(2 - z))^(2))	Hypergeometric2F1[a, b, 2*b, z]=(1 -Divide[1,2]*z)^(- a)* Hypergeometric2F1[Divide[1,2]*a, Divide[1,2]*a +Divide[1,2], b +Divide[1,2], (Divide[z,2 - z])^(2)]	Failure	Failure	Skip	Skip
15.8.E14	${\displaystyle{\displaystyle F\left({a,b\atop 2b};z\right)=\left(1-z\right)^{-\ifrac{a}{2}}F\left({\tfrac{1}{2}a,b-\tfrac{1}{2}a\atop b+\tfrac{1}{2}};\frac{z^{2}}{4z-4}\right)}}$	hypergeom([a, b], [2*b], z)=(1 - z)^(-(a)/(2))* hypergeom([(1)/(2)*a, b -(1)/(2)*a], [b +(1)/(2)], ((z)^(2))/(4*z - 4))	Hypergeometric2F1[a, b, 2*b, z]=(1 - z)^(-Divide[a,2])* Hypergeometric2F1[Divide[1,2]*a, b -Divide[1,2]*a, b +Divide[1,2], Divide[(z)^(2),4*z - 4]]	Failure	Failure	Skip	Successful
15.8.E15	${\displaystyle{\displaystyle F\left({a,b\atop a-b+1};z\right)=(1+z)^{-a}F\left({\frac{1}{2}a,\frac{1}{2}a+\frac{1}{2}\atop a-b+1};\frac{4z}{(1+z)^{2}}\right)}}$	hypergeom([a, b], [a - b + 1], z)=(1 + z)^(- a)* hypergeom([(1)/(2)*a, (1)/(2)*a +(1)/(2)], [a - b + 1], (4*z)/((1 + z)^(2)))	Hypergeometric2F1[a, b, a - b + 1, z]=(1 + z)^(- a)* Hypergeometric2F1[Divide[1,2]*a, Divide[1,2]*a +Divide[1,2], a - b + 1, Divide[4*z,(1 + z)^(2)]]	Failure	Failure	Successful	Successful
15.8.E16	${\displaystyle{\displaystyle F\left({a,b\atop a-b+1};z\right)=(1-z)^{-a}F\left({\frac{1}{2}a,\frac{1}{2}a-b+\frac{1}{2}\atop a-b+1};\frac{-4z}{(1-z)^{2}}\right)}}$	hypergeom([a, b], [a - b + 1], z)=(1 - z)^(- a)* hypergeom([(1)/(2)*a, (1)/(2)*a - b +(1)/(2)], [a - b + 1], (- 4*z)/((1 - z)^(2)))	Hypergeometric2F1[a, b, a - b + 1, z]=(1 - z)^(- a)* Hypergeometric2F1[Divide[1,2]*a, Divide[1,2]*a - b +Divide[1,2], a - b + 1, Divide[- 4*z,(1 - z)^(2)]]	Failure	Failure	Successful	Successful
15.8.E17	${\displaystyle{\displaystyle F\left({a,b\atop\frac{1}{2}(a+b+1)};z\right)=(1-2z)^{-a}F\left({\frac{1}{2}a,\frac{1}{2}a+\frac{1}{2}\atop\frac{1}{2}(a+b+1)};\frac{4z(z-1)}{(1-2z)^{2}}\right)}}$	hypergeom([a, b], [(1)/(2)*(a + b + 1)], z)=(1 - 2*z)^(- a)* hypergeom([(1)/(2)*a, (1)/(2)*a +(1)/(2)], [(1)/(2)*(a + b + 1)], (4*z*(z - 1))/((1 - 2*z)^(2)))	Hypergeometric2F1[a, b, Divide[1,2]*(a + b + 1), z]=(1 - 2*z)^(- a)* Hypergeometric2F1[Divide[1,2]*a, Divide[1,2]*a +Divide[1,2], Divide[1,2]*(a + b + 1), Divide[4*z*(z - 1),(1 - 2*z)^(2)]]	Failure	Failure	Successful	Successful
15.8.E18	${\displaystyle{\displaystyle F\left({a,b\atop\frac{1}{2}(a+b+1)};z\right)=F\left({\frac{1}{2}a,\frac{1}{2}b\atop\frac{1}{2}(a+b+1)};4z(1-z)\right)}}$	hypergeom([a, b], [(1)/(2)*(a + b + 1)], z)= hypergeom([(1)/(2)*a, (1)/(2)*b], [(1)/(2)*(a + b + 1)], 4*z*(1 - z))	Hypergeometric2F1[a, b, Divide[1,2]*(a + b + 1), z]= Hypergeometric2F1[Divide[1,2]*a, Divide[1,2]*b, Divide[1,2]*(a + b + 1), 4*z*(1 - z)]	Failure	Failure	Successful	Successful
15.8.E19	${\displaystyle{\displaystyle F\left({a,1-a\atop c};z\right)=(1-2z)^{1-a-c}(1-z)^{c-1}F\left({\frac{1}{2}(a+c),\frac{1}{2}(a+c-1)\atop c};\frac{4z(z-1)}{(1-2z)^{2}}\right)}}$	hypergeom([a, 1 - a], [c], z)=(1 - 2*z)^(1 - a - c)*(1 - z)^(c - 1)* hypergeom([(1)/(2)*(a + c), (1)/(2)*(a + c - 1)], [c], (4*z*(z - 1))/((1 - 2*z)^(2)))	Hypergeometric2F1[a, 1 - a, c, z]=(1 - 2*z)^(1 - a - c)*(1 - z)^(c - 1)* Hypergeometric2F1[Divide[1,2]*(a + c), Divide[1,2]*(a + c - 1), c, Divide[4*z*(z - 1),(1 - 2*z)^(2)]]	Failure	Failure	Successful	Successful
15.8.E20	${\displaystyle{\displaystyle F\left({a,1-a\atop c};z\right)=(1-z)^{c-1}F\left({\frac{1}{2}(c-a),\frac{1}{2}(a+c-1)\atop c};4z(1-z)\right)}}$	hypergeom([a, 1 - a], [c], z)=(1 - z)^(c - 1)* hypergeom([(1)/(2)*(c - a), (1)/(2)*(a + c - 1)], [c], 4*z*(1 - z))	Hypergeometric2F1[a, 1 - a, c, z]=(1 - z)^(c - 1)* Hypergeometric2F1[Divide[1,2]*(c - a), Divide[1,2]*(a + c - 1), c, 4*z*(1 - z)]	Failure	Failure	Successful	Successful
15.8.E21	${\displaystyle{\displaystyle F\left({a,b\atop a-b+1};z\right)=\left(1+\sqrt{z}\right)^{-2a}F\left({a,a-b+\tfrac{1}{2}\atop 2a-2b+1};\frac{4\sqrt{z}}{(1+\sqrt{z})^{2}}\right)}}$	hypergeom([a, b], [a - b + 1], z)=(1 +sqrt(z))^(- 2*a)* hypergeom([a, a - b +(1)/(2)], [2*a - 2*b + 1], (4*sqrt(z))/((1 +sqrt(z))^(2)))	Hypergeometric2F1[a, b, a - b + 1, z]=(1 +Sqrt[z])^(- 2*a)* Hypergeometric2F1[a, a - b +Divide[1,2], 2*a - 2*b + 1, Divide[4*Sqrt[z],(1 +Sqrt[z])^(2)]]	Failure	Failure	Error	Error
15.8.E22	${\displaystyle{\displaystyle F\left({a,b\atop\tfrac{1}{2}(a+b+1)};z\right)=\left(\frac{\sqrt{1-z^{-1}}-1}{\sqrt{1-z^{-1}}+1}\right)^{a}F\left({a,\tfrac{1}{2}(a+b)\atop a+b};\frac{4\sqrt{1-z^{-1}}}{\left(\sqrt{1-z^{-1}}+1\right)^{2}}\right)}}$	hypergeom([a, b], [(1)/(2)*(a + b + 1)], z)=((sqrt(1 - (z)^(- 1))- 1)/(sqrt(1 - (z)^(- 1))+ 1))^(a)* hypergeom([a, (1)/(2)*(a + b)], [a + b], (4*sqrt(1 - (z)^(- 1)))/((sqrt(1 - (z)^(- 1))+ 1)^(2)))	Hypergeometric2F1[a, b, Divide[1,2]*(a + b + 1), z]=(Divide[Sqrt[1 - (z)^(- 1)]- 1,Sqrt[1 - (z)^(- 1)]+ 1])^(a)* Hypergeometric2F1[a, Divide[1,2]*(a + b), a + b, Divide[4*Sqrt[1 - (z)^(- 1)],(Sqrt[1 - (z)^(- 1)]+ 1)^(2)]]	Failure	Failure	Skip	Successful
15.8.E23	${\displaystyle{\displaystyle F\left({a,1-a\atop c};z\right)=\left(\sqrt{1-z^{-1}}-1\right)^{1-a}\left(\sqrt{1-z^{-1}}+1\right)^{a-2c+1}\left(1-z^{-1}\right)^{c-1}F\left({c-a,c-\tfrac{1}{2}\atop 2c-1};\frac{4\sqrt{1-z^{-1}}}{\left(\sqrt{1-z^{-1}}+1\right)^{2}}\right)}}$	hypergeom([a, 1 - a], [c], z)=(sqrt(1 - (z)^(- 1))- 1)^(1 - a)*(sqrt(1 - (z)^(- 1))+ 1)^(a - 2*c + 1)*(1 - (z)^(- 1))^(c - 1)* hypergeom([c - a, c -(1)/(2)], [2*c - 1], (4*sqrt(1 - (z)^(- 1)))/((sqrt(1 - (z)^(- 1))+ 1)^(2)))	Hypergeometric2F1[a, 1 - a, c, z]=(Sqrt[1 - (z)^(- 1)]- 1)^(1 - a)*(Sqrt[1 - (z)^(- 1)]+ 1)^(a - 2*c + 1)*(1 - (z)^(- 1))^(c - 1)* Hypergeometric2F1[c - a, c -Divide[1,2], 2*c - 1, Divide[4*Sqrt[1 - (z)^(- 1)],(Sqrt[1 - (z)^(- 1)]+ 1)^(2)]]	Failure	Failure	Skip	Successful
15.8.E24	${\displaystyle{\displaystyle F\left({a,b\atop a-b+1};z\right)=(1-z)^{-a}\frac{\Gamma\left(a-b+1\right)\Gamma\left(\tfrac{1}{2}\right)}{\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{2}\right)\Gamma\left(\tfrac{1}{2}a-b+1\right)}F\left({\tfrac{1}{2}a,\tfrac{1}{2}a-b+\tfrac{1}{2}\atop\tfrac{1}{2}};\left(\frac{z+1}{z-1}\right)^{2}\right)+(1+z)(1-z)^{-a-1}\frac{\Gamma\left(a-b+1\right)\Gamma\left(-\tfrac{1}{2}\right)}{\Gamma\left(\tfrac{1}{2}a\right)\Gamma\left(\tfrac{1}{2}a-b+\tfrac{1}{2}\right)}F\left({\tfrac{1}{2}a+\tfrac{1}{2},\tfrac{1}{2}a-b+1\atop\tfrac{3}{2}};\left(\frac{z+1}{z-1}\right)^{2}\right)}}$	hypergeom([a, b], [a - b + 1], z)=(1 - z)^(- a)*(GAMMA(a - b + 1)*GAMMA((1)/(2)))/(GAMMA((1)/(2)*a +(1)/(2))*GAMMA((1)/(2)*a - b + 1))*hypergeom([(1)/(2)*a, (1)/(2)*a - b +(1)/(2)], [(1)/(2)], ((z + 1)/(z - 1))^(2))+(1 + z)*(1 - z)^(- a - 1)*(GAMMA(a - b + 1)*GAMMA(-(1)/(2)))/(GAMMA((1)/(2)*a)*GAMMA((1)/(2)*a - b +(1)/(2)))*hypergeom([(1)/(2)*a +(1)/(2), (1)/(2)*a - b + 1], [(3)/(2)], ((z + 1)/(z - 1))^(2))	Hypergeometric2F1[a, b, a - b + 1, z]=(1 - z)^(- a)*Divide[Gamma[a - b + 1]*Gamma[Divide[1,2]],Gamma[Divide[1,2]*a +Divide[1,2]]*Gamma[Divide[1,2]*a - b + 1]]*Hypergeometric2F1[Divide[1,2]*a, Divide[1,2]*a - b +Divide[1,2], Divide[1,2], (Divide[z + 1,z - 1])^(2)]+(1 + z)*(1 - z)^(- a - 1)*Divide[Gamma[a - b + 1]*Gamma[-Divide[1,2]],Gamma[Divide[1,2]*a]*Gamma[Divide[1,2]*a - b +Divide[1,2]]]*Hypergeometric2F1[Divide[1,2]*a +Divide[1,2], Divide[1,2]*a - b + 1, Divide[3,2], (Divide[z + 1,z - 1])^(2)]	Failure	Failure	Skip	Skip
15.8.E25	${\displaystyle{\displaystyle F\left({a,b\atop\tfrac{1}{2}(a+b+1)};z\right)=\frac{\Gamma\left(\tfrac{1}{2}(a+b+1)\right)\Gamma\left(\tfrac{1}{2}\right)}{\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{2}\right)\Gamma\left(\tfrac{1}{2}b+\tfrac{1}{2}\right)}F\left({\tfrac{1}{2}a,\tfrac{1}{2}b\atop\tfrac{1}{2}};(1-2z)^{2}\right)+(1-2z)\frac{\Gamma\left(\tfrac{1}{2}(a+b+1)\right)\Gamma\left(-\tfrac{1}{2}\right)}{\Gamma\left(\tfrac{1}{2}a\right)\Gamma\left(\tfrac{1}{2}b\right)}F\left({\tfrac{1}{2}a+\tfrac{1}{2},\tfrac{1}{2}b+\tfrac{1}{2}\atop\tfrac{3}{2}};(1-2z)^{2}\right)}}$	hypergeom([a, b], [(1)/(2)*(a + b + 1)], z)=(GAMMA((1)/(2)*(a + b + 1))*GAMMA((1)/(2)))/(GAMMA((1)/(2)*a +(1)/(2))*GAMMA((1)/(2)*b +(1)/(2)))*hypergeom([(1)/(2)*a, (1)/(2)*b], [(1)/(2)], (1 - 2*z)^(2))+(1 - 2*z)*(GAMMA((1)/(2)*(a + b + 1))*GAMMA(-(1)/(2)))/(GAMMA((1)/(2)*a)*GAMMA((1)/(2)*b))*hypergeom([(1)/(2)*a +(1)/(2), (1)/(2)*b +(1)/(2)], [(3)/(2)], (1 - 2*z)^(2))	Hypergeometric2F1[a, b, Divide[1,2]*(a + b + 1), z]=Divide[Gamma[Divide[1,2]*(a + b + 1)]*Gamma[Divide[1,2]],Gamma[Divide[1,2]*a +Divide[1,2]]*Gamma[Divide[1,2]*b +Divide[1,2]]]*Hypergeometric2F1[Divide[1,2]*a, Divide[1,2]*b, Divide[1,2], (1 - 2*z)^(2)]+(1 - 2*z)*Divide[Gamma[Divide[1,2]*(a + b + 1)]*Gamma[-Divide[1,2]],Gamma[Divide[1,2]*a]*Gamma[Divide[1,2]*b]]*Hypergeometric2F1[Divide[1,2]*a +Divide[1,2], Divide[1,2]*b +Divide[1,2], Divide[3,2], (1 - 2*z)^(2)]	Failure	Failure	Skip	Skip
15.8.E26	${\displaystyle{\displaystyle F\left({a,1-a\atop c};z\right)=(1-z)^{c-1}\frac{\Gamma\left(c\right)\Gamma\left(\tfrac{1}{2}\right)}{\Gamma\left(\tfrac{1}{2}(c-a+1)\right)\Gamma\left(\tfrac{1}{2}c+\tfrac{1}{2}a\right)}F\left({\tfrac{1}{2}c-\tfrac{1}{2}a,\tfrac{1}{2}c+\tfrac{1}{2}a-\tfrac{1}{2}\atop\tfrac{1}{2}};(1-2z)^{2}\right)+(1-2z)(1-z)^{c-1}\frac{\Gamma\left(c\right)\Gamma\left(-\tfrac{1}{2}\right)}{\Gamma\left(\tfrac{1}{2}c-\tfrac{1}{2}a\right)\Gamma\left(\tfrac{1}{2}(c+a-1)\right)}F\left({\tfrac{1}{2}c-\tfrac{1}{2}a+\tfrac{1}{2},\tfrac{1}{2}c+\tfrac{1}{2}a\atop\tfrac{3}{2}};(1-2z)^{2}\right)}}$	hypergeom([a, 1 - a], [c], z)=(1 - z)^(c - 1)*(GAMMA(c)*GAMMA((1)/(2)))/(GAMMA((1)/(2)*(c - a + 1))*GAMMA((1)/(2)*c +(1)/(2)*a))*hypergeom([(1)/(2)*c -(1)/(2)*a, (1)/(2)*c +(1)/(2)*a -(1)/(2)], [(1)/(2)], (1 - 2*z)^(2))+(1 - 2*z)*(1 - z)^(c - 1)*(GAMMA(c)*GAMMA(-(1)/(2)))/(GAMMA((1)/(2)*c -(1)/(2)*a)*GAMMA((1)/(2)*(c + a - 1)))*hypergeom([(1)/(2)*c -(1)/(2)*a +(1)/(2), (1)/(2)*c +(1)/(2)*a], [(3)/(2)], (1 - 2*z)^(2))	Hypergeometric2F1[a, 1 - a, c, z]=(1 - z)^(c - 1)*Divide[Gamma[c]*Gamma[Divide[1,2]],Gamma[Divide[1,2]*(c - a + 1)]*Gamma[Divide[1,2]*c +Divide[1,2]*a]]*Hypergeometric2F1[Divide[1,2]*c -Divide[1,2]*a, Divide[1,2]*c +Divide[1,2]*a -Divide[1,2], Divide[1,2], (1 - 2*z)^(2)]+(1 - 2*z)*(1 - z)^(c - 1)*Divide[Gamma[c]*Gamma[-Divide[1,2]],Gamma[Divide[1,2]*c -Divide[1,2]*a]*Gamma[Divide[1,2]*(c + a - 1)]]*Hypergeometric2F1[Divide[1,2]*c -Divide[1,2]*a +Divide[1,2], Divide[1,2]*c +Divide[1,2]*a, Divide[3,2], (1 - 2*z)^(2)]	Failure	Failure	Skip	Successful
15.8.E27	${\displaystyle{\displaystyle\frac{2\Gamma\left(\tfrac{1}{2}\right)\Gamma\left(a+b+\tfrac{1}{2}\right)}{\Gamma\left(a+\tfrac{1}{2}\right)\Gamma\left(b+\tfrac{1}{2}\right)}F\left(a,b;\tfrac{1}{2};z\right)=F\left(2a,2b;a+b+\tfrac{1}{2};\tfrac{1}{2}-\tfrac{1}{2}\sqrt{z}\right)+F\left(2a,2b;a+b+\tfrac{1}{2};\tfrac{1}{2}+\tfrac{1}{2}\sqrt{z}\right)}}$	(2*GAMMA((1)/(2))*GAMMA(a + b +(1)/(2)))/(GAMMA(a +(1)/(2))*GAMMA(b +(1)/(2)))*hypergeom([a, b], [(1)/(2)], z)= hypergeom([2*a, 2*b], [a + b +(1)/(2)], (1)/(2)-(1)/(2)*sqrt(z))+ hypergeom([2*a, 2*b], [a + b +(1)/(2)], (1)/(2)+(1)/(2)*sqrt(z))	Divide[2*Gamma[Divide[1,2]]*Gamma[a + b +Divide[1,2]],Gamma[a +Divide[1,2]]*Gamma[b +Divide[1,2]]]*Hypergeometric2F1[a, b, Divide[1,2], z]= Hypergeometric2F1[2*a, 2*b, a + b +Divide[1,2], Divide[1,2]-Divide[1,2]*Sqrt[z]]+ Hypergeometric2F1[2*a, 2*b, a + b +Divide[1,2], Divide[1,2]+Divide[1,2]*Sqrt[z]]	Failure	Failure	Skip	Skip
15.8.E28	${\displaystyle{\displaystyle\frac{2\sqrt{z}\Gamma\left(-\tfrac{1}{2}\right)\Gamma\left(a+b-\tfrac{1}{2}\right)}{\Gamma\left(a-\tfrac{1}{2}\right)\Gamma\left(b-\tfrac{1}{2}\right)}F\left(a,b;\tfrac{3}{2};z\right)=F\left(2a-1,2b-1;a+b-\tfrac{1}{2};\tfrac{1}{2}-\tfrac{1}{2}\sqrt{z}\right)-F\left(2a-1,2b-1;a+b-\tfrac{1}{2};\tfrac{1}{2}+\tfrac{1}{2}\sqrt{z}\right)}}$	(2*sqrt(z)*GAMMA(-(1)/(2))*GAMMA(a + b -(1)/(2)))/(GAMMA(a -(1)/(2))*GAMMA(b -(1)/(2)))*hypergeom([a, b], [(3)/(2)], z)= hypergeom([2*a - 1, 2*b - 1], [a + b -(1)/(2)], (1)/(2)-(1)/(2)*sqrt(z))- hypergeom([2*a - 1, 2*b - 1], [a + b -(1)/(2)], (1)/(2)+(1)/(2)*sqrt(z))	Divide[2*Sqrt[z]*Gamma[-Divide[1,2]]*Gamma[a + b -Divide[1,2]],Gamma[a -Divide[1,2]]*Gamma[b -Divide[1,2]]]*Hypergeometric2F1[a, b, Divide[3,2], z]= Hypergeometric2F1[2*a - 1, 2*b - 1, a + b -Divide[1,2], Divide[1,2]-Divide[1,2]*Sqrt[z]]- Hypergeometric2F1[2*a - 1, 2*b - 1, a + b -Divide[1,2], Divide[1,2]+Divide[1,2]*Sqrt[z]]	Failure	Failure	Skip	Skip
15.8.E29	${\displaystyle{\displaystyle F\left({a,\tfrac{1}{3}a+\tfrac{1}{3}\atop\tfrac{2}{3}a+\tfrac{2}{3}};z\right)=\left(1+\sqrt{z}\right)^{-2a}\*F\left({a,\tfrac{2}{3}a+\tfrac{1}{6}\atop\tfrac{4}{3}a+\tfrac{1}{3}};\frac{4\sqrt{z}}{(1+\sqrt{z})^{2}}\right)}}$	hypergeom([a, (1)/(3)*a +(1)/(3)], [(2)/(3)*a +(2)/(3)], z)=(1 +sqrt(z))^(- 2*a)* hypergeom([a, (2)/(3)*a +(1)/(6)], [(4)/(3)*a +(1)/(3)], (4*sqrt(z))/((1 +sqrt(z))^(2)))	Hypergeometric2F1[a, Divide[1,3]*a +Divide[1,3], Divide[2,3]*a +Divide[2,3], z]=(1 +Sqrt[z])^(- 2*a)* Hypergeometric2F1[a, Divide[2,3]*a +Divide[1,6], Divide[4,3]*a +Divide[1,3], Divide[4*Sqrt[z],(1 +Sqrt[z])^(2)]]	Failure	Failure	Fail 1.372516024+1.825805270*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)} -.6051524755-3.068921398*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)} .2975793921-.5999898475*I <- {a = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)} .990518232e-1-9.951495097*I <- {a = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Fail Complex[1.372516027919868, 1.8258052682131423] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.6051524788001073, -3.068921398743132] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[0.29757939171102116, -0.5999898475376467] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[0.09905183258496669, -9.951495092476645] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
15.8.E30	${\displaystyle{\displaystyle\left(1-\tfrac{1}{2}z\right)^{-a}F\left({\tfrac{1}{2}a,\tfrac{1}{2}a+\tfrac{1}{2}\atop\tfrac{1}{3}a+\tfrac{5}{6}};\left(\frac{z}{2-z}\right)^{2}\right)=F\left({a,\tfrac{1}{3}a+\tfrac{1}{3}\atop\tfrac{2}{3}a+\tfrac{2}{3}};z\right)}}$	(1 -(1)/(2)*z)^(- a)* hypergeom([(1)/(2)*a, (1)/(2)*a +(1)/(2)], [(1)/(3)*a +(5)/(6)], ((z)/(2 - z))^(2))= hypergeom([a, (1)/(3)*a +(1)/(3)], [(2)/(3)*a +(2)/(3)], z)	(1 -Divide[1,2]*z)^(- a)* Hypergeometric2F1[Divide[1,2]*a, Divide[1,2]*a +Divide[1,2], Divide[1,3]*a +Divide[5,6], (Divide[z,2 - z])^(2)]= Hypergeometric2F1[a, Divide[1,3]*a +Divide[1,3], Divide[2,3]*a +Divide[2,3], z]	Failure	Failure	Successful	Successful
15.8.E30	${\displaystyle{\displaystyle F\left({a,\tfrac{1}{3}a+\tfrac{1}{3}\atop\tfrac{2}{3}a+\tfrac{2}{3}};z\right)=(1+z)^{-a}F\left({\tfrac{1}{2}a,\tfrac{1}{2}a+\tfrac{1}{2}\atop\tfrac{2}{3}a+\tfrac{2}{3}};\frac{4z}{(1+z)^{2}}\right)}}$	hypergeom([a, (1)/(3)*a +(1)/(3)], [(2)/(3)*a +(2)/(3)], z)=(1 + z)^(- a)* hypergeom([(1)/(2)*a, (1)/(2)*a +(1)/(2)], [(2)/(3)*a +(2)/(3)], (4*z)/((1 + z)^(2)))	Hypergeometric2F1[a, Divide[1,3]*a +Divide[1,3], Divide[2,3]*a +Divide[2,3], z]=(1 + z)^(- a)* Hypergeometric2F1[Divide[1,2]*a, Divide[1,2]*a +Divide[1,2], Divide[2,3]*a +Divide[2,3], Divide[4*z,(1 + z)^(2)]]	Failure	Failure	Fail 1.372516027+1.825805270*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)} -.6051524754-3.068921398*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)} .2975793922-.5999898476*I <- {a = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)} .990518493e-1-9.951495087*I <- {a = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Fail Complex[1.372516027919873, 1.825805268213136] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.605152478800107, -3.0689213987431314] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[0.2975793917110212, -0.5999898475376467] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[0.09905183258496181, -9.951495092476698] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
15.8.E31	${\displaystyle{\displaystyle F\left({3a,3a+\frac{1}{2}\atop 4a+\frac{2}{3}};z\right)=\left(1-\tfrac{9}{8}z\right)^{-2a}\*F\left({a,a+\frac{1}{2}\atop 2a+\frac{5}{6}};\frac{27z^{2}(z-1)}{(9z-8)^{2}}\right)}}$	hypergeom([3*a, 3*a +(1)/(2)], [4*a +(2)/(3)], z)=(1 -(9)/(8)*z)^(- 2*a)* hypergeom([a, a +(1)/(2)], [2*a +(5)/(6)], (27*(z)^(2)*(z - 1))/((9*z - 8)^(2)))	Hypergeometric2F1[3*a, 3*a +Divide[1,2], 4*a +Divide[2,3], z]=(1 -Divide[9,8]*z)^(- 2*a)* Hypergeometric2F1[a, a +Divide[1,2], 2*a +Divide[5,6], Divide[27*(z)^(2)*(z - 1),(9*z - 8)^(2)]]	Failure	Failure	Successful	Successful
15.8.E32	${\displaystyle{\displaystyle\frac{\left(1-z^{3}\right)^{a}}{\left(-z\right)^{3a}}\left(\frac{1}{\Gamma\left(a+\frac{2}{3}\right)\Gamma\left(\frac{2}{3}\right)}F\left({a,a+\frac{1}{3}\atop\frac{2}{3}};z^{-3}\right)+\frac{e^{\frac{1}{3}\pi\mathrm{i}}}{z\Gamma\left(a\right)\Gamma\left(\frac{4}{3}\right)}F\left({a+\frac{1}{3},a+\frac{2}{3}\atop\frac{4}{3}};z^{-3}\right)\right)=\frac{3^{\frac{3}{2}a+\frac{1}{2}}e^{\frac{1}{2}a\pi\mathrm{i}}\Gamma\left(a+\frac{1}{3}\right)(1-\zeta)^{a}}{2\pi\Gamma\left(2a+\frac{2}{3}\right)(-\zeta)^{2a}}F\left({a+\frac{1}{3},3a\atop 2a+\frac{2}{3}};\zeta^{-1}\right)}}$	((1 - (z)^(3))^(a))/((- z)^(3*a))*((1)/(GAMMA(a +(2)/(3))*GAMMA((2)/(3)))*hypergeom([a, a +(1)/(3)], [(2)/(3)], (z)^(- 3))+(exp((1)/(3)*Pi*I))/(z*GAMMA(a)*GAMMA((4)/(3)))*hypergeom([a +(1)/(3), a +(2)/(3)], [(4)/(3)], (z)^(- 3)))=((3)^((3)/(2)*a +(1)/(2))* exp((1)/(2)*a*Pi*I)*GAMMA(a +(1)/(3))*(1 - zeta)^(a))/(2*Pi*GAMMA(2*a +(2)/(3))*(- zeta)^(2*a))*hypergeom([a +(1)/(3), 3*a], [2*a +(2)/(3)], (zeta)^(- 1))	Divide[(1 - (z)^(3))^(a),(- z)^(3*a)]*(Divide[1,Gamma[a +Divide[2,3]]*Gamma[Divide[2,3]]]*Hypergeometric2F1[a, a +Divide[1,3], Divide[2,3], (z)^(- 3)]+Divide[Exp[Divide[1,3]*Pi*I],z*Gamma[a]*Gamma[Divide[4,3]]]*Hypergeometric2F1[a +Divide[1,3], a +Divide[2,3], Divide[4,3], (z)^(- 3)])=Divide[(3)^(Divide[3,2]*a +Divide[1,2])* Exp[Divide[1,2]*a*Pi*I]*Gamma[a +Divide[1,3]]*(1 - \[zeta])^(a),2*Pi*Gamma[2*a +Divide[2,3]]*(- \[zeta])^(2*a)]*Hypergeometric2F1[a +Divide[1,3], 3*a, 2*a +Divide[2,3], (\[zeta])^(- 1)]	Failure	Failure	Error	Error
15.8.E33	${\displaystyle{\displaystyle F\left({\frac{1}{3},\frac{2}{3}\atop 1};1-\left(\frac{1-z}{1+2z}\right)^{3}\right)=(1+2z)F\left({\frac{1}{3},\frac{2}{3}\atop 1};z^{3}\right)}}$	hypergeom([(1)/(3), (2)/(3)], [1], 1 -((1 - z)/(1 + 2*z))^(3))=(1 + 2*z)* hypergeom([(1)/(3), (2)/(3)], [1], (z)^(3))	Hypergeometric2F1[Divide[1,3], Divide[2,3], 1, 1 -(Divide[1 - z,1 + 2*z])^(3)]=(1 + 2*z)* Hypergeometric2F1[Divide[1,3], Divide[2,3], 1, (z)^(3)]	Failure	Failure	Fail -.10742540e-1-1.736124843*I <- {z = 2^(1/2)+I*2^(1/2)} -.10742540e-1+1.736124843*I <- {z = 2^(1/2)-I*2^(1/2)} 3.107241801+.4009013497*I <- {z = -2^(1/2)-I*2^(1/2)} 3.107241801-.4009013497*I <- {z = -2^(1/2)+I*2^(1/2)}	Fail Complex[-0.010742539950905128, -1.7361248428967333] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.010742539950905128, 1.7361248428967333] <- {Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[3.107241800778924, 0.40090134898439433] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[3.107241800778924, -0.40090134898439433] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
15.9.E1	${\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\left(x\right)=\frac{{\left(\alpha+1\right)_{n}}}{n!}F\left({-n,n+\alpha+\beta+1\atop\alpha+1};\frac{1-x}{2}\right)}}$	JacobiP(n, alpha, beta, x)=(pochhammer(alpha + 1, n))/(factorial(n))*hypergeom([- n, n + alpha + beta + 1], [alpha + 1], (1 - x)/(2))	JacobiP[n, \[Alpha], \[Beta], x]=Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]*Hypergeometric2F1[- n, n + \[Alpha]+ \[Beta]+ 1, \[Alpha]+ 1, Divide[1 - x,2]]	Successful	Successful	-	-
15.9.E2	${\displaystyle{\displaystyle C^{(\lambda)}_{n}\left(x\right)=\frac{{\left(2\lambda\right)_{n}}}{n!}F\left({-n,n+2\lambda\atop\lambda+\frac{1}{2}};\frac{1-x}{2}\right)}}$	GegenbauerC(n, lambda, x)=(pochhammer(2*lambda, n))/(factorial(n))*hypergeom([- n, n + 2*lambda], [lambda +(1)/(2)], (1 - x)/(2))	GegenbauerC[n, \[Lambda], x]=Divide[Pochhammer[2*\[Lambda], n],(n)!]*Hypergeometric2F1[- n, n + 2*\[Lambda], \[Lambda]+Divide[1,2], Divide[1 - x,2]]	Successful	Successful	-	-
15.9.E3	${\displaystyle{\displaystyle C^{(\lambda)}_{n}\left(x\right)=(2x)^{n}\frac{{\left(\lambda\right)_{n}}}{n!}F\left({-\frac{1}{2}n,\frac{1}{2}(1-n)\atop 1-\lambda-n};\frac{1}{x^{2}}\right)}}$	GegenbauerC(n, lambda, x)=(2*x)^(n)*(pochhammer(lambda, n))/(factorial(n))*hypergeom([-(1)/(2)*n, (1)/(2)*(1 - n)], [1 - lambda - n], (1)/((x)^(2)))	GegenbauerC[n, \[Lambda], x]=(2*x)^(n)*Divide[Pochhammer[\[Lambda], n],(n)!]*Hypergeometric2F1[-Divide[1,2]*n, Divide[1,2]*(1 - n), 1 - \[Lambda]- n, Divide[1,(x)^(2)]]	Failure	Failure	Successful	Successful
15.9.E4	${\displaystyle{\displaystyle C^{(\lambda)}_{n}\left(\cos\theta\right)=e^{n\mathrm{i}\theta}\frac{{\left(\lambda\right)_{n}}}{n!}F\left({-n,\lambda\atop 1-\lambda-n};e^{-2\mathrm{i}\theta}\right)}}$	GegenbauerC(n, lambda, cos(theta))= exp(n*I*theta)*(pochhammer(lambda, n))/(factorial(n))*hypergeom([- n, lambda], [1 - lambda - n], exp(- 2*I*theta))	GegenbauerC[n, \[Lambda], Cos[\[Theta]]]= Exp[n*I*\[Theta]]*Divide[Pochhammer[\[Lambda], n],(n)!]*Hypergeometric2F1[- n, \[Lambda], 1 - \[Lambda]- n, Exp[- 2*I*\[Theta]]]	Failure	Failure	Successful	Successful
15.9.E5	${\displaystyle{\displaystyle T_{n}\left(x\right)=F\left({-n,n\atop\frac{1}{2}};\frac{1-x}{2}\right)}}$	ChebyshevT(n, x)= hypergeom([- n, n], [(1)/(2)], (1 - x)/(2))	ChebyshevT[n, x]= Hypergeometric2F1[- n, n, Divide[1,2], Divide[1 - x,2]]	Successful	Successful	-	-
15.9.E6	${\displaystyle{\displaystyle U_{n}\left(x\right)=(n+1)F\left({-n,n+2\atop\frac{3}{2}};\frac{1-x}{2}\right)}}$	ChebyshevU(n, x)=(n + 1)* hypergeom([- n, n + 2], [(3)/(2)], (1 - x)/(2))	ChebyshevU[n, x]=(n + 1)* Hypergeometric2F1[- n, n + 2, Divide[3,2], Divide[1 - x,2]]	Successful	Failure	-	Successful
15.9.E7	${\displaystyle{\displaystyle P_{n}\left(x\right)=F\left({-n,n+1\atop 1};\frac{1-x}{2}\right)}}$	LegendreP(n, x)= hypergeom([- n, n + 1], [1], (1 - x)/(2))	LegendreP[n, x]= Hypergeometric2F1[- n, n + 1, 1, Divide[1 - x,2]]	Successful	Successful	-	-
15.9.E11	${\displaystyle{\displaystyle\phi^{(\alpha,\beta)}_{\lambda}\left(t\right)=F\left({\tfrac{1}{2}(\alpha+\beta+1-\mathrm{i}\lambda),\tfrac{1}{2}(\alpha+\beta+1+\mathrm{i}\lambda)\atop\alpha+1};-{\sinh^{2}}t\right)}}$	hypergeom([((alpha)+(beta)+1-I*(lambda))/2, ((alpha)+(beta)+1+I*(lambda))], [(alpha)+1], -sinh(t)^2)= hypergeom([(1)/(2)*(alpha + beta + 1 - I*lambda), (1)/(2)*(alpha + beta + 1 + I*lambda)], [alpha + 1], - (sinh(t))^(2))	Error	Failure	Error	Fail 1042.578545-886.9426609*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), lambda = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2)} .4942284159e-3+.1205933063e-2*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), lambda = 2^(1/2)+I*2^(1/2), t = 2^(1/2)-I*2^(1/2)} 1042.578545-886.9426609*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), lambda = 2^(1/2)+I*2^(1/2), t = -2^(1/2)-I*2^(1/2)} .4942284159e-3+.1205933063e-2*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), lambda = 2^(1/2)+I*2^(1/2), t = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	-
15.9.E15	${\displaystyle{\displaystyle C^{(\lambda)}_{\alpha}\left(z\right)=\frac{\Gamma\left(\alpha+2\lambda\right)}{\Gamma\left(2\lambda\right)\Gamma\left(\alpha+1\right)}F\left({-\alpha,\alpha+2\lambda\atop\lambda+\tfrac{1}{2}};\frac{1-z}{2}\right)}}$	GegenbauerC(alpha, lambda, z)=(GAMMA(alpha + 2*lambda))/(GAMMA(2*lambda)*GAMMA(alpha + 1))*hypergeom([- alpha, alpha + 2*lambda], [lambda +(1)/(2)], (1 - z)/(2))	GegenbauerC[\[Alpha], \[Lambda], z]=Divide[Gamma[\[Alpha]+ 2*\[Lambda]],Gamma[2*\[Lambda]]*Gamma[\[Alpha]+ 1]]*Hypergeometric2F1[- \[Alpha], \[Alpha]+ 2*\[Lambda], \[Lambda]+Divide[1,2], Divide[1 - z,2]]	Successful	Successful	-	-
15.9.E16	${\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop 2b};z\right)=\frac{\sqrt{\pi}}{\Gamma\left(b\right)}z^{-b+(\ifrac{1}{2})}(1-z)^{(b-a-(\ifrac{1}{2}))/2}\*P^{-b+(\ifrac{1}{2})}_{a-b-(\ifrac{1}{2})}\left(\frac{2-z}{2\sqrt{1-z}}\right)}}$	hypergeom([a, b], [2*b], z)/GAMMA(2*b)=(sqrt(Pi))/(GAMMA(b))*(z)^(- b +((1)/(2)))*(1 - z)^((b - a -((1)/(2)))/ 2)* LegendreP(a - b -((1)/(2)), - b +((1)/(2)), (2 - z)/(2*sqrt(1 - z)))	Hypergeometric2F1Regularized[a, b, 2*b, z]=Divide[Sqrt[Pi],Gamma[b]]*(z)^(- b +(Divide[1,2]))*(1 - z)^((b - a -(Divide[1,2]))/ 2)* LegendreP[a - b -(Divide[1,2]), - b +(Divide[1,2]), 3, Divide[2 - z,2*Sqrt[1 - z]]]	Failure	Failure	Error	Error
15.9.E17	${\displaystyle{\displaystyle\mathbf{F}\left({a,a+\tfrac{1}{2}\atop c};z\right)=2^{c-1}z^{\ifrac{(1-c)}{2}}(1-z)^{-a+(\ifrac{(c-1)}{2})}\*P^{1-c}_{2a-c}\left(\frac{1}{\sqrt{1-z}}\right)}}$	hypergeom([a, a +(1)/(2)], [c], z)/GAMMA(c)= (2)^(c - 1)* (z)^((1 - c)/(2))*(1 - z)^(- a +((c - 1)/(2)))* LegendreP(2*a - c, 1 - c, (1)/(sqrt(1 - z)))	Hypergeometric2F1Regularized[a, a +Divide[1,2], c, z]= (2)^(c - 1)* (z)^(Divide[1 - c,2])*(1 - z)^(- a +(Divide[c - 1,2]))* LegendreP[2*a - c, 1 - c, 3, Divide[1,Sqrt[1 - z]]]	Failure	Failure	Skip	Successful
15.9.E19	${\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop a-b+1};z\right)=z^{\ifrac{(b-a)}{2}}(1-z)^{-b}\*P^{b-a}_{-b}\left(\frac{1+z}{1-z}\right)}}$	hypergeom([a, b], [a - b + 1], z)/GAMMA(a - b + 1)= (z)^((b - a)/(2))*(1 - z)^(- b)* LegendreP(- b, b - a, (1 + z)/(1 - z))	Hypergeometric2F1Regularized[a, b, a - b + 1, z]= (z)^(Divide[b - a,2])*(1 - z)^(- b)* LegendreP[- b, b - a, 3, Divide[1 + z,1 - z]]	Successful	Failure	-	Successful
15.9.E20	${\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop\tfrac{1}{2}(a+b+1)};z\right)=\left(-z(1-z)\right)^{\ifrac{(1-a-b)}{4}}\*P^{\ifrac{(1-a-b)}{2}}_{\ifrac{(a-b-1)}{2}}\left(1-2z\right)}}$	hypergeom([a, b], [(1)/(2)*(a + b + 1)], z)/GAMMA((1)/(2)*(a + b + 1))=(- z*(1 - z))^((1 - a - b)/(4))* LegendreP((a - b - 1)/(2), (1 - a - b)/(2), 1 - 2*z)	Hypergeometric2F1Regularized[a, b, Divide[1,2]*(a + b + 1), z]=(- z*(1 - z))^(Divide[1 - a - b,4])* LegendreP[Divide[a - b - 1,2], Divide[1 - a - b,2], 3, 1 - 2*z]	Failure	Failure	Skip	Fail Complex[6.0300259512809715, 8.154472102119673] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-17.372827152675953, -0.5381367643934912] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-2.4668594623806452, -1.8721092491501343] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-2.4668594623806426, 1.8721092491501325] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
15.9.E21	${\displaystyle{\displaystyle\mathbf{F}\left({a,1-a\atop c};z\right)=\left(\frac{-z}{1-z}\right)^{\ifrac{(1-c)}{2}}\*P^{1-c}_{-a}\left(1-2z\right)}}$	hypergeom([a, 1 - a], [c], z)/GAMMA(c)=((- z)/(1 - z))^((1 - c)/(2))* LegendreP(- a, 1 - c, 1 - 2*z)	Hypergeometric2F1Regularized[a, 1 - a, c, z]=(Divide[- z,1 - z])^(Divide[1 - c,2])* LegendreP[- a, 1 - c, 3, 1 - 2*z]	Failure	Successful	Skip	-
15.10.E1	${\displaystyle{\displaystyle z(1-z)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(c-(a+b+1)z\right)\frac{\mathrm{d}w}{\mathrm{d}z}-abw=0}}$	z*(1 - z)* diff(w, [z$(2)])+(c -(a + b + 1)*z)* diff(w, z)- a*b*w = 0	z*(1 - z)* D[w, {z, 2}]+(c -(a + b + 1)*z)* D[w, z]- a*b*w = 0	Failure	Failure	Fail 5.656854245-5.656854245*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2)} -5.656854245-5.656854245*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), w = 2^(1/2)-I*2^(1/2)} -5.656854245+5.656854245*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), w = -2^(1/2)-I*2^(1/2)} 5.656854245+5.656854245*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), w = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Fail Complex[5.656854249492381, -5.656854249492381] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-5.656854249492381, -5.656854249492381] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-5.656854249492381, 5.656854249492381] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[5.656854249492381, 5.656854249492381] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
15.10#Ex1	${\displaystyle{\displaystyle f_{1}(z)=F\left({a,b\atop c};z\right)}}$	f[1]*(z)= hypergeom([a, b], [c], z)	Subscript[f, 1]*(z)= Hypergeometric2F1[a, b, c, z]	Failure	Failure	Skip	Skip
15.10#Ex2	${\displaystyle{\displaystyle f_{2}(z)=z^{1-c}F\left({a-c+1,b-c+1\atop 2-c};z\right)}}$	f[2]*(z)= (z)^(1 - c)* hypergeom([a - c + 1, b - c + 1], [2 - c], z)	Subscript[f, 2]*(z)= (z)^(1 - c)* Hypergeometric2F1[a - c + 1, b - c + 1, 2 - c, z]	Failure	Failure	Skip	Skip
15.10#Ex3	${\displaystyle{\displaystyle f_{1}(z)=F\left({a,b\atop a+b+1-c};1-z\right)}}$	f[1]*(z)= hypergeom([a, b], [a + b + 1 - c], 1 - z)	Subscript[f, 1]*(z)= Hypergeometric2F1[a, b, a + b + 1 - c, 1 - z]	Failure	Failure	Skip	Skip
15.10#Ex4	${\displaystyle{\displaystyle f_{2}(z)=(1-z)^{c-a-b}F\left({c-a,c-b\atop c-a-b+1};1-z\right)}}$	f[2]*(z)=(1 - z)^(c - a - b)* hypergeom([c - a, c - b], [c - a - b + 1], 1 - z)	Subscript[f, 2]*(z)=(1 - z)^(c - a - b)* Hypergeometric2F1[c - a, c - b, c - a - b + 1, 1 - z]	Failure	Failure	Skip	Skip
15.10#Ex5	${\displaystyle{\displaystyle f_{1}(z)=z^{-a}F\left({a,a-c+1\atop a-b+1};\frac{1}{z}\right)}}$	f[1]*(z)= (z)^(- a)* hypergeom([a, a - c + 1], [a - b + 1], (1)/(z))	Subscript[f, 1]*(z)= (z)^(- a)* Hypergeometric2F1[a, a - c + 1, a - b + 1, Divide[1,z]]	Failure	Failure	Skip	Skip
15.10#Ex6	${\displaystyle{\displaystyle f_{2}(z)=z^{-b}F\left({b,b-c+1\atop b-a+1};\frac{1}{z}\right)}}$	f[2]*(z)= (z)^(- b)* hypergeom([b, b - c + 1], [b - a + 1], (1)/(z))	Subscript[f, 2]*(z)= (z)^(- b)* Hypergeometric2F1[b, b - c + 1, b - a + 1, Divide[1,z]]	Failure	Failure	Skip	Skip
15.10.E11	${\displaystyle{\displaystyle w_{1}(z)=F\left({a,b\atop c};z\right)}}$	w[1]*(z)= hypergeom([a, b], [c], z)	Subscript[w, 1]*(z)= Hypergeometric2F1[a, b, c, z]	Failure	Failure	Skip	Skip
15.10.E11	${\displaystyle{\displaystyle F\left({a,b\atop c};z\right)=(1-z)^{c-a-b}F\left({c-a,c-b\atop c};z\right)}}$	hypergeom([a, b], [c], z)=(1 - z)^(c - a - b)* hypergeom([c - a, c - b], [c], z)	Hypergeometric2F1[a, b, c, z]=(1 - z)^(c - a - b)* Hypergeometric2F1[c - a, c - b, c, z]	Failure	Successful	Successful	-
15.10.E11	${\displaystyle{\displaystyle(1-z)^{c-a-b}F\left({c-a,c-b\atop c};z\right)=(1-z)^{-a}F\left({a,c-b\atop c};\frac{z}{z-1}\right)}}$	(1 - z)^(c - a - b)* hypergeom([c - a, c - b], [c], z)=(1 - z)^(- a)* hypergeom([a, c - b], [c], (z)/(z - 1))	(1 - z)^(c - a - b)* Hypergeometric2F1[c - a, c - b, c, z]=(1 - z)^(- a)* Hypergeometric2F1[a, c - b, c, Divide[z,z - 1]]	Failure	Failure	Skip	Skip
15.10.E11	${\displaystyle{\displaystyle(1-z)^{-a}F\left({a,c-b\atop c};\frac{z}{z-1}\right)=(1-z)^{-b}F\left({c-a,b\atop c};\frac{z}{z-1}\right)}}$	(1 - z)^(- a)* hypergeom([a, c - b], [c], (z)/(z - 1))=(1 - z)^(- b)* hypergeom([c - a, b], [c], (z)/(z - 1))	(1 - z)^(- a)* Hypergeometric2F1[a, c - b, c, Divide[z,z - 1]]=(1 - z)^(- b)* Hypergeometric2F1[c - a, b, c, Divide[z,z - 1]]	Failure	Failure	Skip	Error
15.10.E12	${\displaystyle{\displaystyle w_{2}(z)={z^{1-c}}F\left({a-c+1,b-c+1\atop 2-c};z\right)}}$	w[2]*(z)=(z)^(1 - c)*hypergeom([a - c + 1, b - c + 1], [2 - c], z)	Subscript[w, 2]*(z)=(z)^(1 - c)*Hypergeometric2F1[a - c + 1, b - c + 1, 2 - c, z]	Failure	Failure	Skip	Error
15.10.E12	${\displaystyle{\displaystyle{z^{1-c}}F\left({a-c+1,b-c+1\atop 2-c};z\right)={z^{1-c}(1-z)^{c-a-b}}\*F\left({1-a,1-b\atop 2-c};z\right)}}$	(z)^(1 - c)*hypergeom([a - c + 1, b - c + 1], [2 - c], z)=(z)^(1 - c)*(1 - z)^(c - a - b)* hypergeom([1 - a, 1 - b], [2 - c], z)	(z)^(1 - c)*Hypergeometric2F1[a - c + 1, b - c + 1, 2 - c, z]=(z)^(1 - c)*(1 - z)^(c - a - b)* Hypergeometric2F1[1 - a, 1 - b, 2 - c, z]	Failure	Successful	Successful	-
15.10.E12	${\displaystyle{\displaystyle{z^{1-c}(1-z)^{c-a-b}}\*F\left({1-a,1-b\atop 2-c};z\right)={z^{1-c}(1-z)^{c-a-1}}\*F\left({a-c+1,1-b\atop 2-c};\frac{z}{z-1}\right)}}$	(z)^(1 - c)*(1 - z)^(c - a - b)* hypergeom([1 - a, 1 - b], [2 - c], z)=(z)^(1 - c)*(1 - z)^(c - a - 1)* hypergeom([a - c + 1, 1 - b], [2 - c], (z)/(z - 1))	(z)^(1 - c)*(1 - z)^(c - a - b)* Hypergeometric2F1[1 - a, 1 - b, 2 - c, z]=(z)^(1 - c)*(1 - z)^(c - a - 1)* Hypergeometric2F1[a - c + 1, 1 - b, 2 - c, Divide[z,z - 1]]	Failure	Failure	Skip	Skip
15.10.E12	${\displaystyle{\displaystyle{z^{1-c}(1-z)^{c-a-1}}\*F\left({a-c+1,1-b\atop 2-c};\frac{z}{z-1}\right)={z^{1-c}(1-z)^{c-b-1}}\*F\left({1-a,b-c+1\atop 2-c};\frac{z}{z-1}\right)}}$	(z)^(1 - c)*(1 - z)^(c - a - 1)* hypergeom([a - c + 1, 1 - b], [2 - c], (z)/(z - 1))=(z)^(1 - c)*(1 - z)^(c - b - 1)* hypergeom([1 - a, b - c + 1], [2 - c], (z)/(z - 1))	(z)^(1 - c)*(1 - z)^(c - a - 1)* Hypergeometric2F1[a - c + 1, 1 - b, 2 - c, Divide[z,z - 1]]=(z)^(1 - c)*(1 - z)^(c - b - 1)* Hypergeometric2F1[1 - a, b - c + 1, 2 - c, Divide[z,z - 1]]	Failure	Failure	Skip	-
15.10.E13	${\displaystyle{\displaystyle w_{3}(z)=F\left({a,b\atop a+b-c+1};1-z\right)}}$	w[3]*(z)= hypergeom([a, b], [a + b - c + 1], 1 - z)	Subscript[w, 3]*(z)= Hypergeometric2F1[a, b, a + b - c + 1, 1 - z]	Failure	Failure	Skip	Error
15.10.E13	${\displaystyle{\displaystyle F\left({a,b\atop a+b-c+1};1-z\right)=z^{1-c}F\left({a-c+1,b-c+1\atop a+b-c+1};1-z\right)}}$	hypergeom([a, b], [a + b - c + 1], 1 - z)= (z)^(1 - c)* hypergeom([a - c + 1, b - c + 1], [a + b - c + 1], 1 - z)	Hypergeometric2F1[a, b, a + b - c + 1, 1 - z]= (z)^(1 - c)* Hypergeometric2F1[a - c + 1, b - c + 1, a + b - c + 1, 1 - z]	Failure	Successful	Successful	-
15.10.E13	${\displaystyle{\displaystyle z^{1-c}F\left({a-c+1,b-c+1\atop a+b-c+1};1-z\right)=z^{-a}F\left({a,a-c+1\atop a+b-c+1};1-\frac{1}{z}\right)}}$	(z)^(1 - c)* hypergeom([a - c + 1, b - c + 1], [a + b - c + 1], 1 - z)= (z)^(- a)* hypergeom([a, a - c + 1], [a + b - c + 1], 1 -(1)/(z))	(z)^(1 - c)* Hypergeometric2F1[a - c + 1, b - c + 1, a + b - c + 1, 1 - z]= (z)^(- a)* Hypergeometric2F1[a, a - c + 1, a + b - c + 1, 1 -Divide[1,z]]	Failure	Failure	Skip	Skip
15.10.E13	${\displaystyle{\displaystyle z^{-a}F\left({a,a-c+1\atop a+b-c+1};1-\frac{1}{z}\right)=z^{-b}F\left({b,b-c+1\atop a+b-c+1};1-\frac{1}{z}\right)}}$	(z)^(- a)* hypergeom([a, a - c + 1], [a + b - c + 1], 1 -(1)/(z))= (z)^(- b)* hypergeom([b, b - c + 1], [a + b - c + 1], 1 -(1)/(z))	(z)^(- a)* Hypergeometric2F1[a, a - c + 1, a + b - c + 1, 1 -Divide[1,z]]= (z)^(- b)* Hypergeometric2F1[b, b - c + 1, a + b - c + 1, 1 -Divide[1,z]]	Failure	Failure	Skip	Error
15.10.E14	${\displaystyle{\displaystyle w_{4}(z)=(1-z)^{c-a-b}F\left({c-a,c-b\atop c-a-b+1};1-z\right)}}$	w[4]*(z)=(1 - z)^(c - a - b)* hypergeom([c - a, c - b], [c - a - b + 1], 1 - z)	Subscript[w, 4]*(z)=(1 - z)^(c - a - b)* Hypergeometric2F1[c - a, c - b, c - a - b + 1, 1 - z]	Failure	Failure	Skip	Error
15.10.E14	${\displaystyle{\displaystyle(1-z)^{c-a-b}F\left({c-a,c-b\atop c-a-b+1};1-z\right)=z^{1-c}(1-z)^{c-a-b}F\left({1-a,1-b\atop c-a-b+1};1-z\right)}}$	(1 - z)^(c - a - b)* hypergeom([c - a, c - b], [c - a - b + 1], 1 - z)= (z)^(1 - c)*(1 - z)^(c - a - b)* hypergeom([1 - a, 1 - b], [c - a - b + 1], 1 - z)	(1 - z)^(c - a - b)* Hypergeometric2F1[c - a, c - b, c - a - b + 1, 1 - z]= (z)^(1 - c)*(1 - z)^(c - a - b)* Hypergeometric2F1[1 - a, 1 - b, c - a - b + 1, 1 - z]	Failure	Successful	Successful	-
15.10.E14	${\displaystyle{\displaystyle z^{1-c}(1-z)^{c-a-b}F\left({1-a,1-b\atop c-a-b+1};1-z\right)=z^{a-c}(1-z)^{c-a-b}F\left({1-a,c-a\atop c-a-b+1};1-\frac{1}{z}\right)}}$	(z)^(1 - c)*(1 - z)^(c - a - b)* hypergeom([1 - a, 1 - b], [c - a - b + 1], 1 - z)= (z)^(a - c)*(1 - z)^(c - a - b)* hypergeom([1 - a, c - a], [c - a - b + 1], 1 -(1)/(z))	(z)^(1 - c)*(1 - z)^(c - a - b)* Hypergeometric2F1[1 - a, 1 - b, c - a - b + 1, 1 - z]= (z)^(a - c)*(1 - z)^(c - a - b)* Hypergeometric2F1[1 - a, c - a, c - a - b + 1, 1 -Divide[1,z]]	Failure	Failure	Skip	Error
15.10.E14	${\displaystyle{\displaystyle z^{a-c}(1-z)^{c-a-b}F\left({1-a,c-a\atop c-a-b+1};1-\frac{1}{z}\right)=z^{b-c}(1-z)^{c-a-b}F\left({1-b,c-b\atop c-a-b+1};1-\frac{1}{z}\right)}}$	(z)^(a - c)*(1 - z)^(c - a - b)* hypergeom([1 - a, c - a], [c - a - b + 1], 1 -(1)/(z))= (z)^(b - c)*(1 - z)^(c - a - b)* hypergeom([1 - b, c - b], [c - a - b + 1], 1 -(1)/(z))	(z)^(a - c)*(1 - z)^(c - a - b)* Hypergeometric2F1[1 - a, c - a, c - a - b + 1, 1 -Divide[1,z]]= (z)^(b - c)*(1 - z)^(c - a - b)* Hypergeometric2F1[1 - b, c - b, c - a - b + 1, 1 -Divide[1,z]]	Failure	Failure	Skip	Error
15.10.E15	${\displaystyle{\displaystyle w_{5}(z)=e^{a\pi\mathrm{i}}z^{-a}\*F\left({a,a-c+1\atop a-b+1};\frac{1}{z}\right)}}$	w[5]*(z)= exp(a*Pi*I)*(z)^(- a)* hypergeom([a, a - c + 1], [a - b + 1], (1)/(z))	Subscript[w, 5]*(z)= Exp[a*Pi*I]*(z)^(- a)* Hypergeometric2F1[a, a - c + 1, a - b + 1, Divide[1,z]]	Failure	Failure	Skip	Error
15.10.E15	${\displaystyle{\displaystyle e^{a\pi\mathrm{i}}z^{-a}\*F\left({a,a-c+1\atop a-b+1};\frac{1}{z}\right)=e^{(c-b)\pi\mathrm{i}}z^{b-c}(1-z)^{c-a-b}\*F\left({1-b,c-b\atop a-b+1};\frac{1}{z}\right)}}$	exp(a*Pi*I)*(z)^(- a)* hypergeom([a, a - c + 1], [a - b + 1], (1)/(z))= exp((c - b)* Pi*I)*(z)^(b - c)*(1 - z)^(c - a - b)* hypergeom([1 - b, c - b], [a - b + 1], (1)/(z))	Exp[a*Pi*I]*(z)^(- a)* Hypergeometric2F1[a, a - c + 1, a - b + 1, Divide[1,z]]= Exp[(c - b)* Pi*I]*(z)^(b - c)*(1 - z)^(c - a - b)* Hypergeometric2F1[1 - b, c - b, a - b + 1, Divide[1,z]]	Failure	Failure	Fail 7.970044489-.2486707840*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)} .2924289380+.3979493992*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)} -111204179.9-19704571.71*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = 2^(1/2)-I*2^(1/2), z = 2^(1/2)-I*2^(1/2)} 37840345.04-37561456.11*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = 2^(1/2)-I*2^(1/2), z = -2^(1/2)-I*2^(1/2)} ... skip entries to safe data	Error
15.10.E15	${\displaystyle{\displaystyle e^{(c-b)\pi\mathrm{i}}z^{b-c}(1-z)^{c-a-b}\*F\left({1-b,c-b\atop a-b+1};\frac{1}{z}\right)=(1-z)^{-a}F\left({a,c-b\atop a-b+1};\frac{1}{1-z}\right)}}$	exp((c - b)* Pi*I)*(z)^(b - c)*(1 - z)^(c - a - b)* hypergeom([1 - b, c - b], [a - b + 1], (1)/(z))=(1 - z)^(- a)* hypergeom([a, c - b], [a - b + 1], (1)/(1 - z))	Exp[(c - b)* Pi*I]*(z)^(b - c)*(1 - z)^(c - a - b)* Hypergeometric2F1[1 - b, c - b, a - b + 1, Divide[1,z]]=(1 - z)^(- a)* Hypergeometric2F1[a, c - b, a - b + 1, Divide[1,1 - z]]	Failure	Failure	Skip	Error
15.10.E15	${\displaystyle{\displaystyle(1-z)^{-a}F\left({a,c-b\atop a-b+1};\frac{1}{1-z}\right)=e^{(c-1)\pi\mathrm{i}}z^{1-c}(1-z)^{c-a-1}\*F\left({1-b,a-c+1\atop a-b+1};\frac{1}{1-z}\right)}}$	(1 - z)^(- a)* hypergeom([a, c - b], [a - b + 1], (1)/(1 - z))= exp((c - 1)* Pi*I)*(z)^(1 - c)*(1 - z)^(c - a - 1)* hypergeom([1 - b, a - c + 1], [a - b + 1], (1)/(1 - z))	(1 - z)^(- a)* Hypergeometric2F1[a, c - b, a - b + 1, Divide[1,1 - z]]= Exp[(c - 1)* Pi*I]*(z)^(1 - c)*(1 - z)^(c - a - 1)* Hypergeometric2F1[1 - b, a - c + 1, a - b + 1, Divide[1,1 - z]]	Failure	Failure	Skip	Error
15.10.E16	${\displaystyle{\displaystyle w_{6}(z)=e^{b\pi\mathrm{i}}z^{-b}F\left({b,b-c+1\atop b-a+1};\frac{1}{z}\right)}}$	w[6]*(z)= exp(b*Pi*I)*(z)^(- b)* hypergeom([b, b - c + 1], [b - a + 1], (1)/(z))	Subscript[w, 6]*(z)= Exp[b*Pi*I]*(z)^(- b)* Hypergeometric2F1[b, b - c + 1, b - a + 1, Divide[1,z]]	Failure	Failure	-	-
15.10.E16	${\displaystyle{\displaystyle e^{b\pi\mathrm{i}}z^{-b}F\left({b,b-c+1\atop b-a+1};\frac{1}{z}\right)=e^{(c-a)\pi\mathrm{i}}z^{a-c}(1-z)^{c-a-b}\*F\left({1-a,c-a\atop b-a+1};\frac{1}{z}\right)}}$	exp(b*Pi*I)*(z)^(- b)* hypergeom([b, b - c + 1], [b - a + 1], (1)/(z))= exp((c - a)* Pi*I)*(z)^(a - c)*(1 - z)^(c - a - b)* hypergeom([1 - a, c - a], [b - a + 1], (1)/(z))	Exp[b*Pi*I]*(z)^(- b)* Hypergeometric2F1[b, b - c + 1, b - a + 1, Divide[1,z]]= Exp[(c - a)* Pi*I]*(z)^(a - c)*(1 - z)^(c - a - b)* Hypergeometric2F1[1 - a, c - a, b - a + 1, Divide[1,z]]	Failure	Failure	-	-
15.10.E16	${\displaystyle{\displaystyle e^{(c-a)\pi\mathrm{i}}z^{a-c}(1-z)^{c-a-b}\*F\left({1-a,c-a\atop b-a+1};\frac{1}{z}\right)=(1-z)^{-b}F\left({b,c-a\atop b-a+1};\frac{1}{1-z}\right)}}$	exp((c - a)* Pi*I)*(z)^(a - c)*(1 - z)^(c - a - b)* hypergeom([1 - a, c - a], [b - a + 1], (1)/(z))=(1 - z)^(- b)* hypergeom([b, c - a], [b - a + 1], (1)/(1 - z))	Exp[(c - a)* Pi*I]*(z)^(a - c)*(1 - z)^(c - a - b)* Hypergeometric2F1[1 - a, c - a, b - a + 1, Divide[1,z]]=(1 - z)^(- b)* Hypergeometric2F1[b, c - a, b - a + 1, Divide[1,1 - z]]	Failure	Failure	-	-
15.10.E16	${\displaystyle{\displaystyle(1-z)^{-b}F\left({b,c-a\atop b-a+1};\frac{1}{1-z}\right)=e^{(c-1)\pi\mathrm{i}}z^{1-c}(1-z)^{c-b-1}\*F\left({1-a,b-c+1\atop b-a+1};\frac{1}{1-z}\right)}}$	(1 - z)^(- b)* hypergeom([b, c - a], [b - a + 1], (1)/(1 - z))= exp((c - 1)* Pi*I)*(z)^(1 - c)*(1 - z)^(c - b - 1)* hypergeom([1 - a, b - c + 1], [b - a + 1], (1)/(1 - z))	(1 - z)^(- b)* Hypergeometric2F1[b, c - a, b - a + 1, Divide[1,1 - z]]= Exp[(c - 1)* Pi*I]*(z)^(1 - c)*(1 - z)^(c - b - 1)* Hypergeometric2F1[1 - a, b - c + 1, b - a + 1, Divide[1,1 - z]]	Failure	Failure	-	-
15.10.E17	${\displaystyle{\displaystyle w_{3}(z)=\frac{\Gamma\left(1-c\right)\Gamma\left(a+b-c+1\right)}{\Gamma\left(a-c+1\right)\Gamma\left(b-c+1\right)}w_{1}(z)+\frac{\Gamma\left(c-1\right)\Gamma\left(a+b-c+1\right)}{\Gamma\left(a\right)\Gamma\left(b\right)}w_{2}(z)}}$	w[3]*(z)=(GAMMA(1 - c)*GAMMA(a + b - c + 1))/(GAMMA(a - c + 1)*GAMMA(b - c + 1))*w[1]*(z)+(GAMMA(c - 1)*GAMMA(a + b - c + 1))/(GAMMA(a)*GAMMA(b))*w[2]*(z)	Subscript[w, 3]*(z)=Divide[Gamma[1 - c]*Gamma[a + b - c + 1],Gamma[a - c + 1]*Gamma[b - c + 1]]*Subscript[w, 1]*(z)+Divide[Gamma[c - 1]*Gamma[a + b - c + 1],Gamma[a]*Gamma[b]]*Subscript[w, 2]*(z)	Failure	Failure	Skip	Error
15.10.E18	${\displaystyle{\displaystyle w_{4}(z)=\frac{\Gamma\left(1-c\right)\Gamma\left(c-a-b+1\right)}{\Gamma\left(1-a\right)\Gamma\left(1-b\right)}w_{1}(z)+\frac{\Gamma\left(c-1\right)\Gamma\left(c-a-b+1\right)}{\Gamma\left(c-a\right)\Gamma\left(c-b\right)}w_{2}(z)}}$	w[4]*(z)=(GAMMA(1 - c)*GAMMA(c - a - b + 1))/(GAMMA(1 - a)*GAMMA(1 - b))*w[1]*(z)+(GAMMA(c - 1)*GAMMA(c - a - b + 1))/(GAMMA(c - a)*GAMMA(c - b))*w[2]*(z)	Subscript[w, 4]*(z)=Divide[Gamma[1 - c]*Gamma[c - a - b + 1],Gamma[1 - a]*Gamma[1 - b]]*Subscript[w, 1]*(z)+Divide[Gamma[c - 1]*Gamma[c - a - b + 1],Gamma[c - a]*Gamma[c - b]]*Subscript[w, 2]*(z)	Failure	Failure	Skip	Error
15.10.E19	${\displaystyle{\displaystyle w_{5}(z)=\frac{\Gamma\left(1-c\right)\Gamma\left(a-b+1\right)}{\Gamma\left(a-c+1\right)\Gamma\left(1-b\right)}w_{1}(z)+e^{(c-1)\pi\mathrm{i}}\frac{\Gamma\left(c-1\right)\Gamma\left(a-b+1\right)}{\Gamma\left(a\right)\Gamma\left(c-b\right)}w_{2}(z)}}$	w[5]*(z)=(GAMMA(1 - c)*GAMMA(a - b + 1))/(GAMMA(a - c + 1)*GAMMA(1 - b))*w[1]*(z)+ exp((c - 1)* Pi*I)*(GAMMA(c - 1)*GAMMA(a - b + 1))/(GAMMA(a)*GAMMA(c - b))*w[2]*(z)	Subscript[w, 5]*(z)=Divide[Gamma[1 - c]*Gamma[a - b + 1],Gamma[a - c + 1]*Gamma[1 - b]]*Subscript[w, 1]*(z)+ Exp[(c - 1)* Pi*I]*Divide[Gamma[c - 1]*Gamma[a - b + 1],Gamma[a]*Gamma[c - b]]*Subscript[w, 2]*(z)	Failure	Failure	Skip	Error
15.10.E20	${\displaystyle{\displaystyle w_{6}(z)=\frac{\Gamma\left(1-c\right)\Gamma\left(b-a+1\right)}{\Gamma\left(b-c+1\right)\Gamma\left(1-a\right)}w_{1}(z)+e^{(c-1)\pi\mathrm{i}}\frac{\Gamma\left(c-1\right)\Gamma\left(b-a+1\right)}{\Gamma\left(b\right)\Gamma\left(c-a\right)}w_{2}(z)}}$	w[6]*(z)=(GAMMA(1 - c)*GAMMA(b - a + 1))/(GAMMA(b - c + 1)*GAMMA(1 - a))*w[1]*(z)+ exp((c - 1)* Pi*I)*(GAMMA(c - 1)*GAMMA(b - a + 1))/(GAMMA(b)*GAMMA(c - a))*w[2]*(z)	Subscript[w, 6]*(z)=Divide[Gamma[1 - c]*Gamma[b - a + 1],Gamma[b - c + 1]*Gamma[1 - a]]*Subscript[w, 1]*(z)+ Exp[(c - 1)* Pi*I]*Divide[Gamma[c - 1]*Gamma[b - a + 1],Gamma[b]*Gamma[c - a]]*Subscript[w, 2]*(z)	Failure	Failure	Skip	Error
15.10.E21	${\displaystyle{\displaystyle w_{1}(z)=\frac{\Gamma\left(c\right)\Gamma\left(c-a-b\right)}{\Gamma\left(c-a\right)\Gamma\left(c-b\right)}w_{3}(z)+\frac{\Gamma\left(c\right)\Gamma\left(a+b-c\right)}{\Gamma\left(a\right)\Gamma\left(b\right)}w_{4}(z)}}$	w[1]*(z)=(GAMMA(c)*GAMMA(c - a - b))/(GAMMA(c - a)*GAMMA(c - b))*w[3]*(z)+(GAMMA(c)*GAMMA(a + b - c))/(GAMMA(a)*GAMMA(b))*w[4]*(z)	Subscript[w, 1]*(z)=Divide[Gamma[c]*Gamma[c - a - b],Gamma[c - a]*Gamma[c - b]]*Subscript[w, 3]*(z)+Divide[Gamma[c]*Gamma[a + b - c],Gamma[a]*Gamma[b]]*Subscript[w, 4]*(z)	Failure	Failure	Skip	Error
15.10.E22	${\displaystyle{\displaystyle w_{2}(z)=\frac{\Gamma\left(2-c\right)\Gamma\left(c-a-b\right)}{\Gamma\left(1-a\right)\Gamma\left(1-b\right)}w_{3}(z)+\frac{\Gamma\left(2-c\right)\Gamma\left(a+b-c\right)}{\Gamma\left(a-c+1\right)\Gamma\left(b-c+1\right)}w_{4}(z)}}$	w[2]*(z)=(GAMMA(2 - c)*GAMMA(c - a - b))/(GAMMA(1 - a)*GAMMA(1 - b))*w[3]*(z)+(GAMMA(2 - c)*GAMMA(a + b - c))/(GAMMA(a - c + 1)*GAMMA(b - c + 1))*w[4]*(z)	Subscript[w, 2]*(z)=Divide[Gamma[2 - c]*Gamma[c - a - b],Gamma[1 - a]*Gamma[1 - b]]*Subscript[w, 3]*(z)+Divide[Gamma[2 - c]*Gamma[a + b - c],Gamma[a - c + 1]*Gamma[b - c + 1]]*Subscript[w, 4]*(z)	Failure	Failure	Skip	Error
15.10.E23	${\displaystyle{\displaystyle w_{5}(z)=e^{a\pi\mathrm{i}}\frac{\Gamma\left(a-b+1\right)\Gamma\left(c-a-b\right)}{\Gamma\left(1-b\right)\Gamma\left(c-b\right)}w_{3}(z)+e^{(c-b)\pi\mathrm{i}}\frac{\Gamma\left(a-b+1\right)\Gamma\left(a+b-c\right)}{\Gamma\left(a\right)\Gamma\left(a-c+1\right)}w_{4}(z)}}$	w[5]*(z)= exp(a*Pi*I)*(GAMMA(a - b + 1)*GAMMA(c - a - b))/(GAMMA(1 - b)*GAMMA(c - b))*w[3]*(z)+ exp((c - b)* Pi*I)*(GAMMA(a - b + 1)*GAMMA(a + b - c))/(GAMMA(a)*GAMMA(a - c + 1))*w[4]*(z)	Subscript[w, 5]*(z)= Exp[a*Pi*I]*Divide[Gamma[a - b + 1]*Gamma[c - a - b],Gamma[1 - b]*Gamma[c - b]]*Subscript[w, 3]*(z)+ Exp[(c - b)* Pi*I]*Divide[Gamma[a - b + 1]*Gamma[a + b - c],Gamma[a]*Gamma[a - c + 1]]*Subscript[w, 4]*(z)	Failure	Failure	Skip	Error
15.10.E24	${\displaystyle{\displaystyle w_{6}(z)=e^{b\pi\mathrm{i}}\frac{\Gamma\left(b-a+1\right)\Gamma\left(c-a-b\right)}{\Gamma\left(1-a\right)\Gamma\left(c-a\right)}w_{3}(z)+e^{(c-a)\pi\mathrm{i}}\frac{\Gamma\left(b-a+1\right)\Gamma\left(a+b-c\right)}{\Gamma\left(b\right)\Gamma\left(b-c+1\right)}w_{4}(z)}}$	w[6]*(z)= exp(b*Pi*I)*(GAMMA(b - a + 1)*GAMMA(c - a - b))/(GAMMA(1 - a)*GAMMA(c - a))*w[3]*(z)+ exp((c - a)* Pi*I)*(GAMMA(b - a + 1)*GAMMA(a + b - c))/(GAMMA(b)*GAMMA(b - c + 1))*w[4]*(z)	Subscript[w, 6]*(z)= Exp[b*Pi*I]*Divide[Gamma[b - a + 1]*Gamma[c - a - b],Gamma[1 - a]*Gamma[c - a]]*Subscript[w, 3]*(z)+ Exp[(c - a)* Pi*I]*Divide[Gamma[b - a + 1]*Gamma[a + b - c],Gamma[b]*Gamma[b - c + 1]]*Subscript[w, 4]*(z)	Failure	Failure	Skip	Error
15.10.E25	${\displaystyle{\displaystyle w_{1}(z)=\frac{\Gamma\left(c\right)\Gamma\left(b-a\right)}{\Gamma\left(b\right)\Gamma\left(c-a\right)}w_{5}(z)+\frac{\Gamma\left(c\right)\Gamma\left(a-b\right)}{\Gamma\left(a\right)\Gamma\left(c-b\right)}w_{6}(z)}}$	w[1]*(z)=(GAMMA(c)*GAMMA(b - a))/(GAMMA(b)*GAMMA(c - a))*w[5]*(z)+(GAMMA(c)*GAMMA(a - b))/(GAMMA(a)*GAMMA(c - b))*w[6]*(z)	Subscript[w, 1]*(z)=Divide[Gamma[c]*Gamma[b - a],Gamma[b]*Gamma[c - a]]*Subscript[w, 5]*(z)+Divide[Gamma[c]*Gamma[a - b],Gamma[a]*Gamma[c - b]]*Subscript[w, 6]*(z)	Failure	Failure	Skip	Error
15.10.E26	${\displaystyle{\displaystyle w_{2}(z)=e^{(1-c)\pi\mathrm{i}}\frac{\Gamma\left(2-c\right)\Gamma\left(b-a\right)}{\Gamma\left(1-a\right)\Gamma\left(b-c+1\right)}w_{5}(z)+e^{(1-c)\pi\mathrm{i}}\frac{\Gamma\left(2-c\right)\Gamma\left(a-b\right)}{\Gamma\left(1-b\right)\Gamma\left(a-c+1\right)}w_{6}(z)}}$	w[2]*(z)= exp((1 - c)* Pi*I)*(GAMMA(2 - c)*GAMMA(b - a))/(GAMMA(1 - a)*GAMMA(b - c + 1))*w[5]*(z)+ exp((1 - c)* Pi*I)*(GAMMA(2 - c)*GAMMA(a - b))/(GAMMA(1 - b)*GAMMA(a - c + 1))*w[6]*(z)	Subscript[w, 2]*(z)= Exp[(1 - c)* Pi*I]*Divide[Gamma[2 - c]*Gamma[b - a],Gamma[1 - a]*Gamma[b - c + 1]]*Subscript[w, 5]*(z)+ Exp[(1 - c)* Pi*I]*Divide[Gamma[2 - c]*Gamma[a - b],Gamma[1 - b]*Gamma[a - c + 1]]*Subscript[w, 6]*(z)	Failure	Failure	Skip	Error
15.10.E27	${\displaystyle{\displaystyle w_{3}(z)=e^{-a\pi\mathrm{i}}\frac{\Gamma\left(a+b-c+1\right)\Gamma\left(b-a\right)}{\Gamma\left(b\right)\Gamma\left(b-c+1\right)}w_{5}(z)+e^{-b\pi\mathrm{i}}\frac{\Gamma\left(a+b-c+1\right)\Gamma\left(a-b\right)}{\Gamma\left(a\right)\Gamma\left(a-c+1\right)}w_{6}(z)}}$	w[3]*(z)= exp(- a*Pi*I)*(GAMMA(a + b - c + 1)*GAMMA(b - a))/(GAMMA(b)*GAMMA(b - c + 1))*w[5]*(z)+ exp(- b*Pi*I)*(GAMMA(a + b - c + 1)*GAMMA(a - b))/(GAMMA(a)*GAMMA(a - c + 1))*w[6]*(z)	Subscript[w, 3]*(z)= Exp[- a*Pi*I]*Divide[Gamma[a + b - c + 1]*Gamma[b - a],Gamma[b]*Gamma[b - c + 1]]*Subscript[w, 5]*(z)+ Exp[- b*Pi*I]*Divide[Gamma[a + b - c + 1]*Gamma[a - b],Gamma[a]*Gamma[a - c + 1]]*Subscript[w, 6]*(z)	Failure	Failure	Skip	Error
15.10.E28	${\displaystyle{\displaystyle w_{4}(z)=e^{(b-c)\pi\mathrm{i}}\frac{\Gamma\left(c-a-b+1\right)\Gamma\left(b-a\right)}{\Gamma\left(1-a\right)\Gamma\left(c-a\right)}w_{5}(z)+e^{(a-c)\pi\mathrm{i}}\frac{\Gamma\left(c-a-b+1\right)\Gamma\left(a-b\right)}{\Gamma\left(1-b\right)\Gamma\left(c-b\right)}w_{6}(z)}}$	w[4]*(z)= exp((b - c)* Pi*I)*(GAMMA(c - a - b + 1)*GAMMA(b - a))/(GAMMA(1 - a)*GAMMA(c - a))*w[5]*(z)+ exp((a - c)* Pi*I)*(GAMMA(c - a - b + 1)*GAMMA(a - b))/(GAMMA(1 - b)*GAMMA(c - b))*w[6]*(z)	Subscript[w, 4]*(z)= Exp[(b - c)* Pi*I]*Divide[Gamma[c - a - b + 1]*Gamma[b - a],Gamma[1 - a]*Gamma[c - a]]*Subscript[w, 5]*(z)+ Exp[(a - c)* Pi*I]*Divide[Gamma[c - a - b + 1]*Gamma[a - b],Gamma[1 - b]*Gamma[c - b]]*Subscript[w, 6]*(z)	Failure	Failure	Skip	Error
15.10.E29	${\displaystyle{\displaystyle w_{1}(z)=e^{b\pi\mathrm{i}}\frac{\Gamma\left(c\right)\Gamma\left(a-c+1\right)}{\Gamma\left(a+b-c+1\right)\Gamma\left(c-b\right)}w_{3}(z)+e^{(b-c)\pi\mathrm{i}}\frac{\Gamma\left(c\right)\Gamma\left(a-c+1\right)}{\Gamma\left(b\right)\Gamma\left(a-b+1\right)}w_{5}(z)}}$	w[1]*(z)= exp(b*Pi*I)*(GAMMA(c)*GAMMA(a - c + 1))/(GAMMA(a + b - c + 1)*GAMMA(c - b))*w[3]*(z)+ exp((b - c)* Pi*I)*(GAMMA(c)*GAMMA(a - c + 1))/(GAMMA(b)*GAMMA(a - b + 1))*w[5]*(z)	Subscript[w, 1]*(z)= Exp[b*Pi*I]*Divide[Gamma[c]*Gamma[a - c + 1],Gamma[a + b - c + 1]*Gamma[c - b]]*Subscript[w, 3]*(z)+ Exp[(b - c)* Pi*I]*Divide[Gamma[c]*Gamma[a - c + 1],Gamma[b]*Gamma[a - b + 1]]*Subscript[w, 5]*(z)	Failure	Failure	Skip	Error
15.10.E30	${\displaystyle{\displaystyle w_{1}(z)=e^{a\pi\mathrm{i}}\frac{\Gamma\left(c\right)\Gamma\left(b-c+1\right)}{\Gamma\left(a+b-c+1\right)\Gamma\left(c-a\right)}w_{3}(z)+e^{(a-c)\pi\mathrm{i}}\frac{\Gamma\left(c\right)\Gamma\left(b-c+1\right)}{\Gamma\left(a\right)\Gamma\left(b-a+1\right)}w_{6}(z)}}$	w[1]*(z)= exp(a*Pi*I)*(GAMMA(c)*GAMMA(b - c + 1))/(GAMMA(a + b - c + 1)*GAMMA(c - a))*w[3]*(z)+ exp((a - c)* Pi*I)*(GAMMA(c)*GAMMA(b - c + 1))/(GAMMA(a)*GAMMA(b - a + 1))*w[6]*(z)	Subscript[w, 1]*(z)= Exp[a*Pi*I]*Divide[Gamma[c]*Gamma[b - c + 1],Gamma[a + b - c + 1]*Gamma[c - a]]*Subscript[w, 3]*(z)+ Exp[(a - c)* Pi*I]*Divide[Gamma[c]*Gamma[b - c + 1],Gamma[a]*Gamma[b - a + 1]]*Subscript[w, 6]*(z)	Failure	Failure	Skip	Error
15.10.E31	${\displaystyle{\displaystyle w_{2}(z)=e^{(b-c+1)\pi\mathrm{i}}\frac{\Gamma\left(2-c\right)\Gamma\left(a\right)}{\Gamma\left(a+b-c+1\right)\Gamma\left(1-b\right)}w_{3}(z)+e^{(b-c)\pi\mathrm{i}}\frac{\Gamma\left(2-c\right)\Gamma\left(a\right)}{\Gamma\left(a-b+1\right)\Gamma\left(b-c+1\right)}w_{5}(z)}}$	w[2]*(z)= exp((b - c + 1)* Pi*I)*(GAMMA(2 - c)*GAMMA(a))/(GAMMA(a + b - c + 1)*GAMMA(1 - b))*w[3]*(z)+ exp((b - c)* Pi*I)*(GAMMA(2 - c)*GAMMA(a))/(GAMMA(a - b + 1)*GAMMA(b - c + 1))*w[5]*(z)	Subscript[w, 2]*(z)= Exp[(b - c + 1)* Pi*I]*Divide[Gamma[2 - c]*Gamma[a],Gamma[a + b - c + 1]*Gamma[1 - b]]*Subscript[w, 3]*(z)+ Exp[(b - c)* Pi*I]*Divide[Gamma[2 - c]*Gamma[a],Gamma[a - b + 1]*Gamma[b - c + 1]]*Subscript[w, 5]*(z)	Failure	Failure	Skip	Error
15.10.E32	${\displaystyle{\displaystyle w_{2}(z)=e^{(a-c+1)\pi\mathrm{i}}\frac{\Gamma\left(2-c\right)\Gamma\left(b\right)}{\Gamma\left(a+b-c+1\right)\Gamma\left(1-a\right)}w_{3}(z)+e^{(a-c)\pi\mathrm{i}}\frac{\Gamma\left(2-c\right)\Gamma\left(b\right)}{\Gamma\left(b-a+1\right)\Gamma\left(a-c+1\right)}w_{6}(z)}}$	w[2]*(z)= exp((a - c + 1)* Pi*I)*(GAMMA(2 - c)*GAMMA(b))/(GAMMA(a + b - c + 1)*GAMMA(1 - a))*w[3]*(z)+ exp((a - c)* Pi*I)*(GAMMA(2 - c)*GAMMA(b))/(GAMMA(b - a + 1)*GAMMA(a - c + 1))*w[6]*(z)	Subscript[w, 2]*(z)= Exp[(a - c + 1)* Pi*I]*Divide[Gamma[2 - c]*Gamma[b],Gamma[a + b - c + 1]*Gamma[1 - a]]*Subscript[w, 3]*(z)+ Exp[(a - c)* Pi*I]*Divide[Gamma[2 - c]*Gamma[b],Gamma[b - a + 1]*Gamma[a - c + 1]]*Subscript[w, 6]*(z)	Failure	Failure	Skip	Error
15.10.E33	${\displaystyle{\displaystyle w_{1}(z)=e^{(c-a)\pi\mathrm{i}}\frac{\Gamma\left(c\right)\Gamma\left(1-b\right)}{\Gamma\left(a\right)\Gamma\left(c-a-b+1\right)}w_{4}(z)+e^{-a\pi\mathrm{i}}\frac{\Gamma\left(c\right)\Gamma\left(1-b\right)}{\Gamma\left(a-b+1\right)\Gamma\left(c-a\right)}w_{5}(z)}}$	w[1]*(z)= exp((c - a)* Pi*I)*(GAMMA(c)*GAMMA(1 - b))/(GAMMA(a)*GAMMA(c - a - b + 1))*w[4]*(z)+ exp(- a*Pi*I)*(GAMMA(c)*GAMMA(1 - b))/(GAMMA(a - b + 1)*GAMMA(c - a))*w[5]*(z)	Subscript[w, 1]*(z)= Exp[(c - a)* Pi*I]*Divide[Gamma[c]*Gamma[1 - b],Gamma[a]*Gamma[c - a - b + 1]]*Subscript[w, 4]*(z)+ Exp[- a*Pi*I]*Divide[Gamma[c]*Gamma[1 - b],Gamma[a - b + 1]*Gamma[c - a]]*Subscript[w, 5]*(z)	Failure	Failure	Skip	Error
15.10.E34	${\displaystyle{\displaystyle w_{1}(z)=e^{(c-b)\pi\mathrm{i}}\frac{\Gamma\left(c\right)\Gamma\left(1-a\right)}{\Gamma\left(b\right)\Gamma\left(c-a-b+1\right)}w_{4}(z)+e^{-b\pi\mathrm{i}}\frac{\Gamma\left(c\right)\Gamma\left(1-a\right)}{\Gamma\left(b-a+1\right)\Gamma\left(c-b\right)}w_{6}(z)}}$	w[1]*(z)= exp((c - b)* Pi*I)*(GAMMA(c)*GAMMA(1 - a))/(GAMMA(b)*GAMMA(c - a - b + 1))*w[4]*(z)+ exp(- b*Pi*I)*(GAMMA(c)*GAMMA(1 - a))/(GAMMA(b - a + 1)*GAMMA(c - b))*w[6]*(z)	Subscript[w, 1]*(z)= Exp[(c - b)* Pi*I]*Divide[Gamma[c]*Gamma[1 - a],Gamma[b]*Gamma[c - a - b + 1]]*Subscript[w, 4]*(z)+ Exp[- b*Pi*I]*Divide[Gamma[c]*Gamma[1 - a],Gamma[b - a + 1]*Gamma[c - b]]*Subscript[w, 6]*(z)	Failure	Failure	Skip	Error
15.10.E35	${\displaystyle{\displaystyle w_{2}(z)=e^{(1-a)\pi\mathrm{i}}\frac{\Gamma\left(2-c\right)\Gamma\left(c-b\right)}{\Gamma\left(a-c+1\right)\Gamma\left(c-a-b+1\right)}w_{4}(z)+e^{-a\pi\mathrm{i}}\frac{\Gamma\left(2-c\right)\Gamma\left(c-b\right)}{\Gamma\left(a-b+1\right)\Gamma\left(1-a\right)}w_{5}(z)}}$	w[2]*(z)= exp((1 - a)* Pi*I)*(GAMMA(2 - c)*GAMMA(c - b))/(GAMMA(a - c + 1)*GAMMA(c - a - b + 1))*w[4]*(z)+ exp(- a*Pi*I)*(GAMMA(2 - c)*GAMMA(c - b))/(GAMMA(a - b + 1)*GAMMA(1 - a))*w[5]*(z)	Subscript[w, 2]*(z)= Exp[(1 - a)* Pi*I]*Divide[Gamma[2 - c]*Gamma[c - b],Gamma[a - c + 1]*Gamma[c - a - b + 1]]*Subscript[w, 4]*(z)+ Exp[- a*Pi*I]*Divide[Gamma[2 - c]*Gamma[c - b],Gamma[a - b + 1]*Gamma[1 - a]]*Subscript[w, 5]*(z)	Failure	Failure	Skip	Error
15.10.E36	${\displaystyle{\displaystyle w_{2}(z)=e^{(1-b)\pi\mathrm{i}}\frac{\Gamma\left(2-c\right)\Gamma\left(c-a\right)}{\Gamma\left(b-c+1\right)\Gamma\left(c-a-b+1\right)}w_{4}(z)+e^{-b\pi\mathrm{i}}\frac{\Gamma\left(2-c\right)\Gamma\left(c-a\right)}{\Gamma\left(b-a+1\right)\Gamma\left(1-b\right)}w_{6}(z)}}$	w[2]*(z)= exp((1 - b)* Pi*I)*(GAMMA(2 - c)*GAMMA(c - a))/(GAMMA(b - c + 1)*GAMMA(c - a - b + 1))*w[4]*(z)+ exp(- b*Pi*I)*(GAMMA(2 - c)*GAMMA(c - a))/(GAMMA(b - a + 1)*GAMMA(1 - b))*w[6]*(z)	Subscript[w, 2]*(z)= Exp[(1 - b)* Pi*I]*Divide[Gamma[2 - c]*Gamma[c - a],Gamma[b - c + 1]*Gamma[c - a - b + 1]]*Subscript[w, 4]*(z)+ Exp[- b*Pi*I]*Divide[Gamma[2 - c]*Gamma[c - a],Gamma[b - a + 1]*Gamma[1 - b]]*Subscript[w, 6]*(z)	Failure	Failure	Skip	Error
15.12.E1	${\displaystyle{\displaystyle\alpha_{+}=\operatorname{arctan}\left(\frac{\operatorname{ph}z-\operatorname{ph}\left(1-z\right)-\pi}{\ln|1-z^{-1}|}\right)}}$	alpha[+]= arctan((argument(z)- argument(1 - z)- Pi)/(ln(abs(1 - (z)^(- 1)))))	Subscript[\[Alpha], +]= ArcTan[Divide[Arg[z]- Arg[1 - z]- Pi,Log[Abs[1 - (z)^(- 1)]]]]	Error	Failure	-	Error
15.12.E1	${\displaystyle{\displaystyle\alpha_{-}=\operatorname{arctan}\left(\frac{\operatorname{ph}z-\operatorname{ph}\left(1-z\right)+\pi}{\ln|1-z^{-1}|}\right)}}$	alpha[-]= arctan((argument(z)- argument(1 - z)+ Pi)/(ln(abs(1 - (z)^(- 1)))))	Subscript[\[Alpha], -]= ArcTan[Divide[Arg[z]- Arg[1 - z]+ Pi,Log[Abs[1 - (z)^(- 1)]]]]	Error	Failure	-	Error
15.12.E6	${\displaystyle{\displaystyle\zeta=\operatorname{arccosh}z}}$	zeta = arccosh(z)	\[zeta]= ArcCosh[z]	Failure	Failure	Fail .22188930e-1+.5671060437*I <- {z = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)+I*2^(1/2)} .22188930e-1-2.261321080*I <- {z = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)-I*2^(1/2)} -2.806238194-2.261321080*I <- {z = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)-I*2^(1/2)} -2.806238194+.5671060437*I <- {z = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Error
15.12.E8	${\displaystyle{\displaystyle\alpha=\left(-2\ln\left(1-\left(\frac{z-1}{z+1}\right)^{2}\right)\right)^{1/2}}}$	alpha =(- 2*ln(1 -((z - 1)/(z + 1))^(2)))^(1/ 2)	\[Alpha]=(- 2*Log[1 -(Divide[z - 1,z + 1])^(2)])^(1/ 2)	Failure	Failure	Fail .9106437259+.8692893105*I <- {alpha = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)} .9106437259+1.959137814*I <- {alpha = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)} .6547904750+3.198787193*I <- {alpha = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)} .6547904750-.370360069*I <- {alpha = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Error
15.12.E10	${\displaystyle{\displaystyle\zeta=\operatorname{arccosh}\left(\tfrac{1}{4}z-1\right)}}$	zeta = arccosh((1)/(4)*z - 1)	\[zeta]= ArcCosh[Divide[1,4]*z - 1]	Failure	Failure	Fail .9885072570-.790108118*I <- {z = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)+I*2^(1/2)} .9885072570-3.618535242*I <- {z = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)-I*2^(1/2)} -1.839919867-3.618535242*I <- {z = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)-I*2^(1/2)} -1.839919867-.790108118*I <- {z = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Error
15.12.E11	${\displaystyle{\displaystyle\beta=\left(-\frac{3}{2}\zeta+\frac{9}{4}\ln\left(\frac{2+e^{\zeta}}{2+e^{-\zeta}}\right)\right)^{1/3}}}$	beta =(-(3)/(2)*zeta +(9)/(4)*ln((2 + exp(zeta))/(2 + exp(- zeta))))^(1/ 3)	\[Beta]=(-Divide[3,2]*\[zeta]+Divide[9,4]*Log[Divide[2 + Exp[\[zeta]],2 + Exp[- \[zeta]]]])^(1/ 3)	Failure	Failure	Fail .8286036743+.9438951834*I <- {beta = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)+I*2^(1/2)} .8286036743+1.884531941*I <- {beta = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)-I*2^(1/2)} .7141009552+1.686207411*I <- {beta = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)-I*2^(1/2)} .7141009552+1.142219713*I <- {beta = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Error
15.13.E1	${\displaystyle{\displaystyle N(a,b,c)=\begin{cases}0,&a}}\)\@add@PDF@RDFa@triples\end{document}\end{cases}$	N*(a , b , c)=	N*(a , b , c)=	Error	Failure	-	Error
15.13.E1	${\displaystyle{\displaystyle\begin{cases}0,&a>0,\\ \left\lfloor-a\right\rfloor+\tfrac{1}{2}(1+S),&a}}\)\@add@PDF@RDFa@triples\end{document}\end{cases}$			Error	Failure	-	Error
15.13.E1	${\displaystyle{\displaystyle 0,\\ \left\lfloor-a\right\rfloor+\tfrac{1}{2}(1+S),&a<0,c-a}}$	0 , floor(- a)+(1)/(2)*(1 + S),	0 , Floor[- a]+Divide[1,2]*(1 + S),	Error	Failure	-	Error
15.14.E1	${\displaystyle{\displaystyle\int_{0}^{\infty}x^{s-1}\mathbf{F}\left({a,b\atop c};-x\right)\mathrm{d}x=\frac{\Gamma\left(s\right)\Gamma\left(a-s\right)\Gamma\left(b-s\right)}{\Gamma\left(a\right)\Gamma\left(b\right)\Gamma\left(c-s\right)}}}$	int((x)^(s - 1)* hypergeom([a, b], [c], - x)/GAMMA(c), x = 0..infinity)=(GAMMA(s)*GAMMA(a - s)*GAMMA(b - s))/(GAMMA(a)*GAMMA(b)*GAMMA(c - s))	Integrate[(x)^(s - 1)* Hypergeometric2F1Regularized[a, b, c, - x], {x, 0, Infinity}]=Divide[Gamma[s]*Gamma[a - s]*Gamma[b - s],Gamma[a]*Gamma[b]*Gamma[c - s]]	Successful	Failure	-	Error
15.15.E1	${\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop c};\frac{1}{z}\right)=\left(1-\frac{z_{0}}{z}\right)^{-a}\sum_{s=0}^{\infty}\frac{(a)_{s}}{s!}\*\mathbf{F}\left({-s,b\atop c};\frac{1}{z_{0}}\right)\left(1-\frac{z}{z_{0}}\right)^{-s}}}$	hypergeom([a, b], [c], (1)/(z))/GAMMA(c)=(1 -(z[0])/(z))^(- a)* sum((a[s])/(factorial(s))* hypergeom([- s, b], [c], (1)/(z[0]))/GAMMA(c)*(1 -(z)/(z[0]))^(- s), s = 0..infinity)	Hypergeometric2F1Regularized[a, b, c, Divide[1,z]]=(1 -Divide[Subscript[z, 0],z])^(- a)* Sum[Divide[Subscript[a, s],(s)!]* Hypergeometric2F1Regularized[- s, b, c, Divide[1,Subscript[z, 0]]]*(1 -Divide[z,Subscript[z, 0]])^(- s), {s, 0, Infinity}]	Failure	Failure	Skip	Error
15.16.E1	${\displaystyle{\displaystyle F\left({a,b\atop c-\frac{1}{2}};z\right)F\left({c-a,c-b\atop c+\frac{1}{2}};z\right)=\sum_{s=0}^{\infty}\frac{{\left(c\right)_{s}}}{{\left(c+\frac{1}{2}\right)_{s}}}A_{s}z^{s}}}$	hypergeom([a, b], [c -(1)/(2)], z)*hypergeom([c - a, c - b], [c +(1)/(2)], z)= sum((pochhammer(c, s))/(pochhammer(c +(1)/(2), s))*A[s]*(z)^(s), s = 0..infinity)	Hypergeometric2F1[a, b, c -Divide[1,2], z]*Hypergeometric2F1[c - a, c - b, c +Divide[1,2], z]= Sum[Divide[Pochhammer[c, s],Pochhammer[c +Divide[1,2], s]]*Subscript[A, s]*(z)^(s), {s, 0, Infinity}]	Failure	Failure	Skip	Error
15.16.E2	${\displaystyle{\displaystyle(1-z)^{a+b-c}F\left(2a,2b;2c-1;z\right)=\sum_{s=0}^{\infty}A_{s}z^{s}}}$	(1 - z)^(a + b - c)* hypergeom([2*a, 2*b], [2*c - 1], z)= sum(A[s]*(z)^(s), s = 0..infinity)	(1 - z)^(a + b - c)* Hypergeometric2F1[2*a, 2*b, 2*c - 1, z]= Sum[Subscript[A, s]*(z)^(s), {s, 0, Infinity}]	Failure	Failure	Skip	Error
15.16.E3	${\displaystyle{\displaystyle F\left({a,b\atop c};z\right)F\left({a,b\atop c};\zeta\right)=\sum_{s=0}^{\infty}\frac{{\left(a\right)_{s}}{\left(b\right)_{s}}{\left(c-a\right)_{s}}{\left(c-b\right)_{s}}}{{\left(c\right)_{s}}{\left(c\right)_{2s}}s!}\left(z\zeta\right)^{s}F\left({a+s,b+s\atop c+2s};z+\zeta-z\zeta\right)}}$	hypergeom([a, b], [c], z)*hypergeom([a, b], [c], zeta)= sum((pochhammer(a, s)*pochhammer(b, s)*pochhammer(c - a, s)*pochhammer(c - b, s))/(pochhammer(c, s)*pochhammer(c, 2*s)*factorial(s))*(z*zeta)^(s)* hypergeom([a + s, b + s], [c + 2*s], z + zeta - z*zeta), s = 0..infinity)	Hypergeometric2F1[a, b, c, z]*Hypergeometric2F1[a, b, c, \[zeta]]= Sum[Divide[Pochhammer[a, s]*Pochhammer[b, s]*Pochhammer[c - a, s]*Pochhammer[c - b, s],Pochhammer[c, s]*Pochhammer[c, 2*s]*(s)!]*(z*\[zeta])^(s)* Hypergeometric2F1[a + s, b + s, c + 2*s, z + \[zeta]- z*\[zeta]], {s, 0, Infinity}]	Failure	Failure	Skip	Error
15.16.E4	${\displaystyle{\displaystyle F\left({a,b\atop c};z\right)F\left({-a,-b\atop-c};z\right)+\frac{ab(a-c)(b-c)}{c^{2}(1-c^{2})}z^{2}F\left({1+a,1+b\atop 2+c};z\right)F\left({1-a,1-b\atop 2-c};z\right)=1}}$	hypergeom([a, b], [c], z)*hypergeom([- a, - b], [- c], z)+(a*b*(a - c)*(b - c))/((c)^(2)*(1 - (c)^(2)))*(z)^(2)* hypergeom([1 + a, 1 + b], [2 + c], z)*hypergeom([1 - a, 1 - b], [2 - c], z)= 1	Hypergeometric2F1[a, b, c, z]*Hypergeometric2F1[- a, - b, - c, z]+Divide[a*b*(a - c)*(b - c),(c)^(2)*(1 - (c)^(2))]*(z)^(2)* Hypergeometric2F1[1 + a, 1 + b, 2 + c, z]*Hypergeometric2F1[1 - a, 1 - b, 2 - c, z]= 1	Failure	Failure	Successful	Error
15.16.E5	${\displaystyle{\displaystyle F\left({\frac{1}{2}+\lambda,-\frac{1}{2}-\nu\atop 1+\lambda+\mu};z\right)F\left({\frac{1}{2}-\lambda,\frac{1}{2}+\nu\atop 1+\nu+\mu};1-z\right)+F\left({\frac{1}{2}+\lambda,\frac{1}{2}-\nu\atop 1+\lambda+\mu};z\right)F\left({-\frac{1}{2}-\lambda,\frac{1}{2}+\nu\atop 1+\nu+\mu};1-z\right)-F\left({\frac{1}{2}+\lambda,\frac{1}{2}-\nu\atop 1+\lambda+\mu};z\right)F\left({\frac{1}{2}-\lambda,\frac{1}{2}+\nu\atop 1+\nu+\mu};1-z\right)=\frac{\Gamma\left(1+\lambda+\mu\right)\Gamma\left(1+\nu+\mu\right)}{\Gamma\left(\lambda+\mu+\nu+\frac{3}{2}\right)\Gamma\left(\frac{1}{2}+\nu\right)}}}$	hypergeom([(1)/(2)+ lambda, -(1)/(2)- nu], [1 + lambda + mu], z)*hypergeom([(1)/(2)- lambda, (1)/(2)+ nu], [1 + nu + mu], 1 - z)+ hypergeom([(1)/(2)+ lambda, (1)/(2)- nu], [1 + lambda + mu], z)*hypergeom([-(1)/(2)- lambda, (1)/(2)+ nu], [1 + nu + mu], 1 - z)- hypergeom([(1)/(2)+ lambda, (1)/(2)- nu], [1 + lambda + mu], z)*hypergeom([(1)/(2)- lambda, (1)/(2)+ nu], [1 + nu + mu], 1 - z)=(GAMMA(1 + lambda + mu)*GAMMA(1 + nu + mu))/(GAMMA(lambda + mu + nu +(3)/(2))*GAMMA((1)/(2)+ nu))	Hypergeometric2F1[Divide[1,2]+ \[Lambda], -Divide[1,2]- \[Nu], 1 + \[Lambda]+ \[Mu], z]*Hypergeometric2F1[Divide[1,2]- \[Lambda], Divide[1,2]+ \[Nu], 1 + \[Nu]+ \[Mu], 1 - z]+ Hypergeometric2F1[Divide[1,2]+ \[Lambda], Divide[1,2]- \[Nu], 1 + \[Lambda]+ \[Mu], z]*Hypergeometric2F1[-Divide[1,2]- \[Lambda], Divide[1,2]+ \[Nu], 1 + \[Nu]+ \[Mu], 1 - z]- Hypergeometric2F1[Divide[1,2]+ \[Lambda], Divide[1,2]- \[Nu], 1 + \[Lambda]+ \[Mu], z]*Hypergeometric2F1[Divide[1,2]- \[Lambda], Divide[1,2]+ \[Nu], 1 + \[Nu]+ \[Mu], 1 - z]=Divide[Gamma[1 + \[Lambda]+ \[Mu]]*Gamma[1 + \[Nu]+ \[Mu]],Gamma[\[Lambda]+ \[Mu]+ \[Nu]+Divide[3,2]]*Gamma[Divide[1,2]+ \[Nu]]]	Failure	Failure	Skip	Error