Results of Generalized Hypergeometric Functions and Meijer G-Function

DLMF Formula Maple Mathematica Symbolic
Maple
Symbolic
Mathematica
Numeric
Maple
Numeric
Mathematica
16.2.E3 ${\displaystyle{\displaystyle{{}_{p+1}F_{q}}\left({-m,\mathbf{a}\atop\mathbf{b}% };z\right)=\frac{{\left(\mathbf{a}\right)_{m}}(-z)^{m}}{{\left(\mathbf{b}% \right)_{m}}}{{}_{q+1}F_{p}}\left({-m,1-m-\mathbf{b}\atop 1-m-\mathbf{a}};% \frac{(-1)^{p+q}}{z}\right)}}$ hypergeom([- m , a], [b], z)=(pochhammer(a, m)*(- z)^(m))/(pochhammer(b, m))*hypergeom([- m , 1 - m - b], [1 - m - a], ((- 1)^(p + q))/(z)) HypergeometricPFQ[{- m , a}, {b}, z]=Divide[Pochhammer[a, m]*(- z)^(m),Pochhammer[b, m]]*HypergeometricPFQ[{- m , 1 - m - b}, {1 - m - a}, Divide[(- 1)^(p + q),z]] Failure Failure Skip Skip
16.2.E4 ${\displaystyle{\displaystyle\sum_{k=0}^{m}\frac{{\left(\mathbf{a}\right)_{k}}}% {{\left(\mathbf{b}\right)_{k}}}\frac{z^{k}}{k!}=\frac{{\left(\mathbf{a}\right)% _{m}}z^{m}}{{\left(\mathbf{b}\right)_{m}}m!}{{}_{q+2}F_{p}}\left({-m,1,1-m-% \mathbf{b}\atop 1-m-\mathbf{a}};\frac{(-1)^{p+q+1}}{z}\right)}}$ sum((pochhammer(a, k))/(pochhammer(b, k))*((z)^(k))/(factorial(k)), k = 0..m)=(pochhammer(a, m)*(z)^(m))/(pochhammer(b, m)*factorial(m))*hypergeom([- m , 1 , 1 - m - b], [1 - m - a], ((- 1)^(p + q + 1))/(z)) Sum[Divide[Pochhammer[a, k],Pochhammer[b, k]]*Divide[(z)^(k),(k)!], {k, 0, m}]=Divide[Pochhammer[a, m]*(z)^(m),Pochhammer[b, m]*(m)!]*HypergeometricPFQ[{- m , 1 , 1 - m - b}, {1 - m - a}, Divide[(- 1)^(p + q + 1),z]] Failure Failure Skip Skip
16.3.E5 ${\displaystyle{\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}=% z^{n}\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}z^{n}}}$ (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
-0.-3.999999998*I <- {z = 2^(1/2)+I*2^(1/2), n = 2}
28.28427122-28.28427122*I <- {z = 2^(1/2)+I*2^(1/2), n = 3}
-0.+3.999999998*I <- {z = 2^(1/2)-I*2^(1/2), n = 2}
28.28427122+28.28427122*I <- {z = 2^(1/2)-I*2^(1/2), n = 3}
... skip entries to safe data
Fail
Complex[0.0, -4.0] <- {Rule[n, 2], 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[0.0, 4.0] <- {Rule[n, 2], 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]]]]}
... skip entries to safe data
16.3.E6 ${\displaystyle{\displaystyle z{{}_{0}F_{1}}\left(-;b+1;z\right)+b(b-1){{}_{0}F% _{1}}\left(-;b;z\right)-b(b-1){{}_{0}F_{1}}\left(-;b-1;z\right)=0}}$ z*hypergeom([-], [b + 1], z)+ b*(b - 1)* hypergeom([-], [b], z)- b*(b - 1)* hypergeom([-], [b - 1], z)= 0 z*HypergeometricPFQ[{-}, {b + 1}, z]+ b*(b - 1)* HypergeometricPFQ[{-}, {b}, z]- b*(b - 1)* HypergeometricPFQ[{-}, {b - 1}, z]= 0 Error Failure - Error
16.3.E7 ${\displaystyle{\displaystyle{{}_{3}F_{2}}\left({a_{1}+2,a_{2},a_{3}\atop b_{1}% ,b_{2}};z\right)a_{1}(a_{1}+1)(1-z)+{{}_{3}F_{2}}\left({a_{1}+1,a_{2},a_{3}% \atop b_{1},b_{2}};z\right)a_{1}\left(b_{1}+b_{2}-3a_{1}-2+z(2a_{1}-a_{2}-a_{3% }+1)\right)+{{}_{3}F_{2}}\left({a_{1},a_{2},a_{3}\atop b_{1},b_{2}};z\right)% \left((2a_{1}-b_{1})(2a_{1}-b_{2})+a_{1}-a_{1}^{2}-z(a_{1}-a_{2})(a_{1}-a_{3})% \right)-{{}_{3}F_{2}}\left({a_{1}-1,a_{2},a_{3}\atop b_{1},b_{2}};z\right)(a_{% 1}-b_{1})(a_{1}-b_{2})=0}}$ hypergeom([a[1]+ 2 , a[2], a[3]], [b[1], b[2]], z)*a[1]*(a[1]+ 1)*(1 - z)+ hypergeom([a[1]+ 1 , a[2], a[3]], [b[1], b[2]], z)*a[1]*(b[1]+ b[2]- 3*a[1]- 2 + z*(2*a[1]- a[2]- a[3]+ 1))+ ((2*a[1]- b[1])*(2*a[1]- b[2])+ a[1]- a(a[1])^(2)- z*(a[1]- a[2])*(a[1]- a[3]))- hypergeom([a[1]- 1 , a[2], a[3]], [b[1], b[2]], z)*(a[1]- b[1])*(a[1]- b[2])= 0 HypergeometricPFQ[{Subscript[a, 1]+ 2 , Subscript[a, 2], Subscript[a, 3]}, {Subscript[b, 1], Subscript[b, 2]}, z]*Subscript[a, 1]*(Subscript[a, 1]+ 1)*(1 - z)+ HypergeometricPFQ[{Subscript[a, 1]+ 1 , Subscript[a, 2], Subscript[a, 3]}, {Subscript[b, 1], Subscript[b, 2]}, z]*Subscript[a, 1]*(Subscript[b, 1]+ Subscript[b, 2]- 3*Subscript[a, 1]- 2 + z*(2*Subscript[a, 1]- Subscript[a, 2]- Subscript[a, 3]+ 1))+ ((2*Subscript[a, 1]- Subscript[b, 1])*(2*Subscript[a, 1]- Subscript[b, 2])+ Subscript[a, 1]- a(Subscript[a, 1])^(2)- z*(Subscript[a, 1]- Subscript[a, 2])*(Subscript[a, 1]- Subscript[a, 3]))- HypergeometricPFQ[{Subscript[a, 1]- 1 , Subscript[a, 2], Subscript[a, 3]}, {Subscript[b, 1], Subscript[b, 2]}, z]*(Subscript[a, 1]- Subscript[b, 1])*(Subscript[a, 1]- Subscript[b, 2])= 0 Failure Failure Skip Skip
16.4.E3 ${\displaystyle{\displaystyle{{}_{3}F_{2}}\left({-n,a,b\atop c,d};1\right)=% \frac{{\left(c-a\right)_{n}}{\left(c-b\right)_{n}}}{{\left(c\right)_{n}}{\left% (c-a-b\right)_{n}}}}}$ hypergeom([- n , a , b], [c , d], 1)=(pochhammer(c - a, n)*pochhammer(c - b, n))/(pochhammer(c, n)*pochhammer(c - a - b, n)) HypergeometricPFQ[{- n , a , b}, {c , d}, 1]=Divide[Pochhammer[c - a, n]*Pochhammer[c - b, n],Pochhammer[c, n]*Pochhammer[c - a - b, n]] Failure Failure Skip Skip
16.4.E4 ${\displaystyle{\displaystyle{{}_{3}F_{2}}\left({a,b,c\atop a-b+1,a-c+1};1% \right)=\frac{\Gamma\left(\frac{1}{2}a+1\right)\Gamma\left(a-b+1\right)\Gamma% \left(a-c+1\right)\Gamma\left(\frac{1}{2}a-b-c+1\right)}{\Gamma\left(a+1\right% )\Gamma\left(\frac{1}{2}a-b+1\right)\Gamma\left(\frac{1}{2}a-c+1\right)\Gamma% \left(a-b-c+1\right)}}}$ hypergeom([a , b , c], [a - b + 1 , a - c + 1], 1)=(GAMMA((1)/(2)*a + 1)*GAMMA(a - b + 1)*GAMMA(a - c + 1)*GAMMA((1)/(2)*a - b - c + 1))/(GAMMA(a + 1)*GAMMA((1)/(2)*a - b + 1)*GAMMA((1)/(2)*a - c + 1)*GAMMA(a - b - c + 1)) HypergeometricPFQ[{a , b , c}, {a - b + 1 , a - c + 1}, 1]=Divide[Gamma[Divide[1,2]*a + 1]*Gamma[a - b + 1]*Gamma[a - c + 1]*Gamma[Divide[1,2]*a - b - c + 1],Gamma[a + 1]*Gamma[Divide[1,2]*a - b + 1]*Gamma[Divide[1,2]*a - c + 1]*Gamma[a - b - c + 1]] Successful Successful - -
16.4.E6 ${\displaystyle{\displaystyle{{}_{3}F_{2}}\left({a,b,c\atop\frac{1}{2}(a+b+1),2% c};1\right)=\frac{\Gamma\left(\frac{1}{2}\right)\Gamma\left(c+\frac{1}{2}% \right)\Gamma\left(\frac{1}{2}(a+b+1)\right)\Gamma\left(c+\frac{1}{2}(1-a-b)% \right)}{\Gamma\left(\frac{1}{2}(a+1)\right)\Gamma\left(\frac{1}{2}(b+1)\right% )\Gamma\left(c+\frac{1}{2}(1-a)\right)\Gamma\left(c+\frac{1}{2}(1-b)\right)}}}$ hypergeom([a , b , c], [(1)/(2)*(a + b + 1), 2*c], 1)=(GAMMA((1)/(2))*GAMMA(c +(1)/(2))*GAMMA((1)/(2)*(a + b + 1))*GAMMA(c +(1)/(2)*(1 - a - b)))/(GAMMA((1)/(2)*(a + 1))*GAMMA((1)/(2)*(b + 1))*GAMMA(c +(1)/(2)*(1 - a))*GAMMA(c +(1)/(2)*(1 - b))) HypergeometricPFQ[{a , b , c}, {Divide[1,2]*(a + b + 1), 2*c}, 1]=Divide[Gamma[Divide[1,2]]*Gamma[c +Divide[1,2]]*Gamma[Divide[1,2]*(a + b + 1)]*Gamma[c +Divide[1,2]*(1 - a - b)],Gamma[Divide[1,2]*(a + 1)]*Gamma[Divide[1,2]*(b + 1)]*Gamma[c +Divide[1,2]*(1 - a)]*Gamma[c +Divide[1,2]*(1 - b)]] Successful Failure - Skip
16.4.E7 ${\displaystyle{\displaystyle{{}_{3}F_{2}}\left({a,1-a,c\atop d,2c-d+1};1\right% )=\frac{\pi\Gamma\left(d\right)\Gamma\left(2c-d+1\right)2^{1-2c}}{\Gamma\left(% c+\frac{1}{2}(a-d+1)\right)\Gamma\left(c+1-\frac{1}{2}(a+d)\right)\Gamma\left(% \frac{1}{2}(a+d)\right)\Gamma\left(\frac{1}{2}(d-a+1)\right)}}}$ hypergeom([a , 1 - a , c], [d , 2*c - d + 1], 1)=(Pi*GAMMA(d)*GAMMA(2*c - d + 1)*(2)^(1 - 2*c))/(GAMMA(c +(1)/(2)*(a - d + 1))*GAMMA(c + 1 -(1)/(2)*(a + d))*GAMMA((1)/(2)*(a + d))*GAMMA((1)/(2)*(d - a + 1))) HypergeometricPFQ[{a , 1 - a , c}, {d , 2*c - d + 1}, 1]=Divide[Pi*Gamma[d]*Gamma[2*c - d + 1]*(2)^(1 - 2*c),Gamma[c +Divide[1,2]*(a - d + 1)]*Gamma[c + 1 -Divide[1,2]*(a + d)]*Gamma[Divide[1,2]*(a + d)]*Gamma[Divide[1,2]*(d - a + 1)]] Successful Successful - -
16.4.E8 ${\displaystyle{\displaystyle{{}_{3}F_{2}}\left({-n,a,1-a\atop d,1-d-2n};1% \right)=\frac{{\left(\frac{1}{2}(a+d)\right)_{n}}{\left(\frac{1}{2}(d-a+1)% \right)_{n}}}{{\left(\frac{1}{2}d\right)_{n}}{\left(\frac{1}{2}(d+1)\right)_{n% }}}}}$ hypergeom([- n , a , 1 - a], [d , 1 - d - 2*n], 1)=(pochhammer((1)/(2)*(a + d), n)*pochhammer((1)/(2)*(d - a + 1), n))/(pochhammer((1)/(2)*d, n)*pochhammer((1)/(2)*(d + 1), n)) HypergeometricPFQ[{- n , a , 1 - a}, {d , 1 - d - 2*n}, 1]=Divide[Pochhammer[Divide[1,2]*(a + d), n]*Pochhammer[Divide[1,2]*(d - a + 1), n],Pochhammer[Divide[1,2]*d, n]*Pochhammer[Divide[1,2]*(d + 1), n]] Failure Failure Skip
Fail
Complex[0.6167812573081491, -0.36130209551358583] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[d, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 1]}
Complex[0.5345650901276873, -0.45277145278729625] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[d, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 2]}
Complex[0.49947205392907457, -0.49663470546035027] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[d, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 3]}
Complex[1.2312873319809505, 0.8850107130731889] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[d, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[n, 1]}
... skip entries to safe data
16.4.E9 ${\displaystyle{\displaystyle{{}_{5}F_{4}}\left({a,\frac{1}{2}a+1,b,c,d\atop% \frac{1}{2}a,a-b+1,a-c+1,a-d+1};1\right)=\frac{\Gamma\left(a-b+1\right)\Gamma% \left(a-c+1\right)\Gamma\left(a-d+1\right)\Gamma\left(a-b-c-d+1\right)}{\Gamma% \left(a+1\right)\Gamma\left(a-b-c+1\right)\Gamma\left(a-b-d+1\right)\Gamma% \left(a-c-d+1\right)}}}$ hypergeom([a ,(1)/(2)*a + 1 , b , c , d], [(1)/(2)*a , a - b + 1 , a - c + 1 , a - d + 1], 1)=(GAMMA(a - b + 1)*GAMMA(a - c + 1)*GAMMA(a - d + 1)*GAMMA(a - b - c - d + 1))/(GAMMA(a + 1)*GAMMA(a - b - c + 1)*GAMMA(a - b - d + 1)*GAMMA(a - c - d + 1)) HypergeometricPFQ[{a ,Divide[1,2]*a + 1 , b , c , d}, {Divide[1,2]*a , a - b + 1 , a - c + 1 , a - d + 1}, 1]=Divide[Gamma[a - b + 1]*Gamma[a - c + 1]*Gamma[a - d + 1]*Gamma[a - b - c - d + 1],Gamma[a + 1]*Gamma[a - b - c + 1]*Gamma[a - b - d + 1]*Gamma[a - c - d + 1]] Failure Failure Skip Skip
16.4.E10 ${\displaystyle{\displaystyle{{}_{7}F_{6}}\left({a,\frac{1}{2}a+1,b,c,d,f,-n% \atop\frac{1}{2}a,a-b+1,a-c+1,a-d+1,a-f+1,a+n+1};1\right)=\frac{{\left(a+1% \right)_{n}}{\left(a-b-c+1\right)_{n}}{\left(a-b-d+1\right)_{n}}{\left(a-c-d+1% \right)_{n}}}{{\left(a-b+1\right)_{n}}{\left(a-c+1\right)_{n}}{\left(a-d+1% \right)_{n}}{\left(a-b-c-d+1\right)_{n}}}}}$ hypergeom([a ,(1)/(2)*a + 1 , b , c , d , f , - n], [(1)/(2)*a , a - b + 1 , a - c + 1 , a - d + 1 , a - f + 1 , a + n + 1], 1)=(pochhammer(a + 1, n)*pochhammer(a - b - c + 1, n)*pochhammer(a - b - d + 1, n)*pochhammer(a - c - d + 1, n))/(pochhammer(a - b + 1, n)*pochhammer(a - c + 1, n)*pochhammer(a - d + 1, n)*pochhammer(a - b - c - d + 1, n)) HypergeometricPFQ[{a ,Divide[1,2]*a + 1 , b , c , d , f , - n}, {Divide[1,2]*a , a - b + 1 , a - c + 1 , a - d + 1 , a - f + 1 , a + n + 1}, 1]=Divide[Pochhammer[a + 1, n]*Pochhammer[a - b - c + 1, n]*Pochhammer[a - b - d + 1, n]*Pochhammer[a - c - d + 1, n],Pochhammer[a - b + 1, n]*Pochhammer[a - c + 1, n]*Pochhammer[a - d + 1, n]*Pochhammer[a - b - c - d + 1, n]] Failure Failure Skip Skip
16.4.E11 ${\displaystyle{\displaystyle{{}_{3}F_{2}}\left({a,b,c\atop d,e};1\right)=\frac% {\Gamma\left(e\right)\Gamma\left(d+e-a-b-c\right)}{\Gamma\left(e-a\right)% \Gamma\left(d+e-b-c\right)}{{}_{3}F_{2}}\left({a,d-b,d-c\atop d,d+e-b-c};1% \right)}}$ hypergeom([a , b , c], [d , e], 1)=(GAMMA(e)*GAMMA(d + e - a - b - c))/(GAMMA(e - a)*GAMMA(d + e - b - c))*hypergeom([a , d - b , d - c], [d , d + e - b - c], 1) HypergeometricPFQ[{a , b , c}, {d , e}, 1]=Divide[Gamma[e]*Gamma[d + e - a - b - c],Gamma[e - a]*Gamma[d + e - b - c]]*HypergeometricPFQ[{a , d - b , d - c}, {d , d + e - b - c}, 1] Failure Failure Skip Skip
16.4.E12 ${\displaystyle{\displaystyle(a-d)(b-d)(c-d)\left({{}_{3}F_{2}}\left({a,b,c% \atop d+1,e};1\right)-{{}_{3}F_{2}}\left({a,b,c\atop d,e};1\right)\right)+abc{% {}_{3}F_{2}}\left({a,b,c\atop d,e};1\right)=d(d-1)(a+b+c-d-e+1)\left({{}_{3}F_% {2}}\left({a,b,c\atop d,e};1\right)-{{}_{3}F_{2}}\left({a,b,c\atop d-1,e};1% \right)\right)}}$ (a - d)*(b - d)*(c - d)*(hypergeom([a , b , c], [d + 1 , e], 1)- hypergeom([a , b , c], [d , e], 1))+ a*b*c*hypergeom([a , b , c], [d , e], 1)= d*(d - 1)*(a + b + c - d - e + 1)*(hypergeom([a , b , c], [d , e], 1)- hypergeom([a , b , c], [d - 1 , e], 1)) (a - d)*(b - d)*(c - d)*(HypergeometricPFQ[{a , b , c}, {d + 1 , e}, 1]- HypergeometricPFQ[{a , b , c}, {d , e}, 1])+ a*b*c*HypergeometricPFQ[{a , b , c}, {d , e}, 1]= d*(d - 1)*(a + b + c - d - e + 1)*(HypergeometricPFQ[{a , b , c}, {d , e}, 1]- HypergeometricPFQ[{a , b , c}, {d - 1 , e}, 1]) Failure Failure Skip Skip
16.4.E13 ${\displaystyle{\displaystyle{{}_{3}F_{2}}\left({a,b,c\atop d,e};1\right)=% \dfrac{c(e-a)}{de}{{}_{3}F_{2}}\left({a,b+1,c+1\atop d+1,e+1};1\right)+\dfrac{% d-c}{d}{{}_{3}F_{2}}\left({a,b+1,c\atop d+1,e};1\right)}}$ hypergeom([a , b , c], [d , e], 1)=(c*(e - a))/(d*e)*hypergeom([a , b + 1 , c + 1], [d + 1 , e + 1], 1)+(d - c)/(d)*hypergeom([a , b + 1 , c], [d + 1 , e], 1) HypergeometricPFQ[{a , b , c}, {d , e}, 1]=Divide[c*(e - a),d*e]*HypergeometricPFQ[{a , b + 1 , c + 1}, {d + 1 , e + 1}, 1]+Divide[d - c,d]*HypergeometricPFQ[{a , b + 1 , c}, {d + 1 , e}, 1] Failure Failure Skip Skip
16.4.E14 ${\displaystyle{\displaystyle{{}_{4}F_{3}}\left({-n,a,b,c\atop d,e,f};1\right)=% \frac{{\left(e-a\right)_{n}}{\left(f-a\right)_{n}}}{{\left(e\right)_{n}}{\left% (f\right)_{n}}}{{}_{4}F_{3}}\left({-n,a,d-b,d-c\atop d,a-e-n+1,a-f-n+1};1% \right)}}$ hypergeom([- n , a , b , c], [d , e , f], 1)=(pochhammer(e - a, n)*pochhammer(f - a, n))/(pochhammer(e, n)*pochhammer(f, n))*hypergeom([- n , a , d - b , d - c], [d , a - e - n + 1 , a - f - n + 1], 1) HypergeometricPFQ[{- n , a , b , c}, {d , e , f}, 1]=Divide[Pochhammer[e - a, n]*Pochhammer[f - a, n],Pochhammer[e, n]*Pochhammer[f, n]]*HypergeometricPFQ[{- n , a , d - b , d - c}, {d , a - e - n + 1 , a - f - n + 1}, 1] Failure Failure Skip Skip
16.4.E15 ${\displaystyle{\displaystyle{{}_{7}F_{6}}\left({a,\frac{1}{2}a+1,b,c,d,e,f% \atop\frac{1}{2}a,a-b+1,a-c+1,a-d+1,a-e+1,a-f+1};1\right)=\frac{\Gamma\left(a-% d+1\right)\Gamma\left(a-e+1\right)\Gamma\left(a-f+1\right)\Gamma\left(a-d-e-f+% 1\right)}{\Gamma\left(a+1\right)\Gamma\left(a-d-e+1\right)\Gamma\left(a-d-f+1% \right)\Gamma\left(a-e-f+1\right)}{{}_{4}F_{3}}\left({a-b-c+1,d,e,f\atop a-b+1% ,a-c+1,d+e+f-a};1\right)}}$ hypergeom([a ,(1)/(2)*a + 1 , b , c , d , e , f], [(1)/(2)*a , a - b + 1 , a - c + 1 , a - d + 1 , a - e + 1 , a - f + 1], 1)=(GAMMA(a - d + 1)*GAMMA(a - e + 1)*GAMMA(a - f + 1)*GAMMA(a - d - e - f + 1))/(GAMMA(a + 1)*GAMMA(a - d - e + 1)*GAMMA(a - d - f + 1)*GAMMA(a - e - f + 1))*hypergeom([a - b - c + 1 , d , e , f], [a - b + 1 , a - c + 1 , d + e + f - a], 1) HypergeometricPFQ[{a ,Divide[1,2]*a + 1 , b , c , d , e , f}, {Divide[1,2]*a , a - b + 1 , a - c + 1 , a - d + 1 , a - e + 1 , a - f + 1}, 1]=Divide[Gamma[a - d + 1]*Gamma[a - e + 1]*Gamma[a - f + 1]*Gamma[a - d - e - f + 1],Gamma[a + 1]*Gamma[a - d - e + 1]*Gamma[a - d - f + 1]*Gamma[a - e - f + 1]]*HypergeometricPFQ[{a - b - c + 1 , d , e , f}, {a - b + 1 , a - c + 1 , d + e + f - a}, 1] Failure Failure Skip Skip
16.6.E1 ${\displaystyle{\displaystyle{{}_{3}F_{2}}\left({a,b,c\atop a-b+1,a-c+1};z% \right)=(1-z)^{-a}{{}_{3}F_{2}}\left({a-b-c+1,\frac{1}{2}a,\frac{1}{2}(a+1)% \atop a-b+1,a-c+1};\frac{-4z}{(1-z)^{2}}\right)}}$ hypergeom([a , b , c], [a - b + 1 , a - c + 1], z)=(1 - z)^(- a)* hypergeom([a - b - c + 1 ,(1)/(2)*a ,(1)/(2)*(a + 1)], [a - b + 1 , a - c + 1], (- 4*z)/((1 - z)^(2))) HypergeometricPFQ[{a , b , c}, {a - b + 1 , a - c + 1}, z]=(1 - z)^(- a)* HypergeometricPFQ[{a - b - c + 1 ,Divide[1,2]*a ,Divide[1,2]*(a + 1)}, {a - b + 1 , a - c + 1}, Divide[- 4*z,(1 - z)^(2)]] Failure Failure Skip Skip
16.6.E2 ${\displaystyle{\displaystyle{{}_{3}F_{2}}\left({a,2b-a-1,2-2b+a\atop b,a-b+% \frac{3}{2}};\frac{z}{4}\right)=(1-z)^{-a}{{}_{3}F_{2}}\left({\frac{1}{3}a,% \frac{1}{3}a+\frac{1}{3},\frac{1}{3}a+\frac{2}{3}\atop b,a-b+\frac{3}{2}};% \frac{-27z}{4(1-z)^{3}}\right)}}$ hypergeom([a , 2*b - a - 1 , 2 - 2*b + a], [b , a - b +(3)/(2)], (z)/(4))=(1 - z)^(- a)* hypergeom([(1)/(3)*a ,(1)/(3)*a +(1)/(3),(1)/(3)*a +(2)/(3)], [b , a - b +(3)/(2)], (- 27*z)/(4*(1 - z)^(3))) HypergeometricPFQ[{a , 2*b - a - 1 , 2 - 2*b + a}, {b , a - b +Divide[3,2]}, Divide[z,4]]=(1 - z)^(- a)* HypergeometricPFQ[{Divide[1,3]*a ,Divide[1,3]*a +Divide[1,3],Divide[1,3]*a +Divide[2,3]}, {b , a - b +Divide[3,2]}, Divide[- 27*z,4*(1 - z)^(3)]] Failure Failure
Fail
1.264484429+.9040719052*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)}
6.675662159-3.231621754*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)}
.9005931042+.1301294103*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)}
.4136818818+.2979732613*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)}
... skip entries to safe data
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16.12.E1 ${\displaystyle{\displaystyle{{}_{0}F_{1}}\left(-;a;z\right){{}_{0}F_{1}}\left(% -;b;z\right)={{}_{2}F_{3}}\left({\frac{1}{2}(a+b),\frac{1}{2}(a+b-1)\atop a,b,% a+b-1};4z\right)}}$ hypergeom([-], [a], z)*hypergeom([-], [b], z)= hypergeom([(1)/(2)*(a + b),(1)/(2)*(a + b - 1)], [a , b , a + b - 1], 4*z) HypergeometricPFQ[{-}, {a}, z]*HypergeometricPFQ[{-}, {b}, z]= HypergeometricPFQ[{Divide[1,2]*(a + b),Divide[1,2]*(a + b - 1)}, {a , b , a + b - 1}, 4*z] Error Failure - Error
16.12.E2 ${\displaystyle{\displaystyle\left({{}_{2}F_{1}}\left({a,b\atop a+b+\frac{1}{2}% };z\right)\right)^{2}={{}_{3}F_{2}}\left({2a,2b,a+b\atop a+b+\frac{1}{2},2a+2b% };z\right)}}$ (hypergeom([a , b], [a + b +(1)/(2)], z))^(2)= hypergeom([2*a , 2*b , a + b], [a + b +(1)/(2), 2*a + 2*b], z) (HypergeometricPFQ[{a , b}, {a + b +Divide[1,2]}, z])^(2)= HypergeometricPFQ[{2*a , 2*b , a + b}, {a + b +Divide[1,2], 2*a + 2*b}, z] Failure Failure
Fail
-19187.29656+12814.21944*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)}
.2505536537-.2824002444*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)}
8.286487590-21.96358656*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)}
160.2876416+891.1260277*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)}
... skip entries to safe data
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16.12.E3 ${\displaystyle{\displaystyle\left({{}_{2}F_{1}}\left({a,b\atop c};z\right)% \right)^{2}=\sum_{k=0}^{\infty}\frac{{\left(2a\right)_{k}}{\left(2b\right)_{k}% }{\left(c-\frac{1}{2}\right)_{k}}}{{\left(c\right)_{k}}{\left(2c-1\right)_{k}}% k!}{{}_{4}F_{3}}\left({-\frac{1}{2}k,\frac{1}{2}(1-k),a+b-c+\frac{1}{2},\frac{% 1}{2}\atop a+\frac{1}{2},b+\frac{1}{2},\frac{3}{2}-k-c};1\right)z^{k}}}$ (hypergeom([a , b], [c], z))^(2)= sum((pochhammer(2*a, k)*pochhammer(2*b, k)*pochhammer(c -(1)/(2), k))/(pochhammer(c, k)*pochhammer(2*c - 1, k)*factorial(k))*hypergeom([-(1)/(2)*k ,(1)/(2)*(1 - k), a + b - c +(1)/(2),(1)/(2)], [a +(1)/(2), b +(1)/(2),(3)/(2)- k - c], 1)*(z)^(k), k = 0..infinity) (HypergeometricPFQ[{a , b}, {c}, z])^(2)= Sum[Divide[Pochhammer[2*a, k]*Pochhammer[2*b, k]*Pochhammer[c -Divide[1,2], k],Pochhammer[c, k]*Pochhammer[2*c - 1, k]*(k)!]*HypergeometricPFQ[{-Divide[1,2]*k ,Divide[1,2]*(1 - k), a + b - c +Divide[1,2],Divide[1,2]}, {a +Divide[1,2], b +Divide[1,2],Divide[3,2]- k - c}, 1]*(z)^(k), {k, 0, Infinity}] Failure Failure Skip Skip
16.16.E5 ${\displaystyle{\displaystyle{F_{3}}\left(\alpha,\gamma-\alpha;\beta,\gamma-% \beta;\gamma;x,y\right)=(1-y)^{\alpha+\beta-\gamma}{{}_{2}F_{1}}\left({\alpha,% \beta\atop\gamma};x+y-xy\right)}}$ Error AppellF[3, , \[Alpha], \[Gamma]- \[Alpha], \[Beta], \[Gamma]- \[Beta]]*\[Gamma]*x*y =(1 - y)^(\[Alpha]+ \[Beta]- \[Gamma])* HypergeometricPFQ[{\[Alpha], \[Beta]}, {\[Gamma]}, x + y - x*y] Error Failure - Skip
16.16.E6 ${\displaystyle{\displaystyle{F_{4}}\left(\alpha,\beta;\gamma,\alpha+\beta-% \gamma+1;x(1-y),y(1-x)\right)={{}_{2}F_{1}}\left({\alpha,\beta\atop\gamma};x% \right){{}_{2}F_{1}}\left({\alpha,\beta\atop\alpha+\beta-\gamma+1};y\right)}}$ Error AppellF[4, , \[Alpha], \[Beta], \[Gamma], \[Alpha]+ \[Beta]- \[Gamma]+ 1]*x*(1 - y)*y*(1 - x)= HypergeometricPFQ[{\[Alpha], \[Beta]}, {\[Gamma]}, x]*HypergeometricPFQ[{\[Alpha], \[Beta]}, {\[Alpha]+ \[Beta]- \[Gamma]+ 1}, y] Error Failure - Skip
16.23.E1 ${\displaystyle{\displaystyle{{}_{3}F_{2}}\left({-n,n+\alpha+2,\frac{1}{2}(% \alpha+1)\atop\alpha+1,\frac{1}{2}(\alpha+3)};x\right)>0}}$ hypergeom([- n , n + alpha + 2 ,(1)/(2)*(alpha + 1)], [alpha + 1 ,(1)/(2)*(alpha + 3)], x)> 0 HypergeometricPFQ[{- n , n + \[Alpha]+ 2 ,Divide[1,2]*(\[Alpha]+ 1)}, {\[Alpha]+ 1 ,Divide[1,2]*(\[Alpha]+ 3)}, x]> 0 Failure Failure Skip Successful