# Results of Hypergeometric Function

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DLMF Formula Maple Mathematica Symbolic
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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 ${\displaystyle{\displaystyle c(c-1)(z-1)F\left(a,b;c-1;z\right)+{c\left(c-1-(2% c-a-b-1)z\right)}F\left(a,b;c;z\right)+(c-a)(c-b)zF\left(a,b;c+1;z\right)=0}}$ 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