Hypergeometric Function - 16.2 Definition and Analytic Properties

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
16.2.E3	${\displaystyle{\displaystyle{{}_{p+1}F_{q}}\left({-m,\mathbf{a}\atop\mathbf{b}};z\right)=\frac{{\left(\mathbf{a}\right)_{m}}(-z)^{m}}{{\left(\mathbf{b}\right)_{m}}}{{}_{q+1}F_{p}}\left({-m,1-m-\mathbf{b}\atop 1-m-\mathbf{a}};\frac{(-1)^{p+q}}{z}\right)}}$ \genhyperF{p+1}{q}@@{-m,\mathbf{a}}{\mathbf{b}}{z} = \frac{\Pochhammersym{\mathbf{a}}{m}(-z)^{m}}{\Pochhammersym{\mathbf{b}}{m}}\genhyperF{q+1}{p}@@{-m,1-m-\mathbf{b}}{1-m-\mathbf{a}}{\frac{(-1)^{p+q}}{z}}		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	Failed [258 / 300] Result: .9712138727+.322304453e-1*I Test Values: {a = -3/2, b = -3/2, p = 1/2*3^(1/2)+1/2*I, q = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, m = 1} Result: -.6497511671-1.025183062*I Test Values: {a = -3/2, b = -3/2, p = 1/2*3^(1/2)+1/2*I, q = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, m = 2} ... skip entries to safe data	Failed [258 / 300] Result: Complex[0.9712138727144691, 0.032230445352325054] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[m, 1], Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.6497511667213578, -1.0251830622105054] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[m, 2], Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
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_{k=0}^{m}\frac{\Pochhammersym{\mathbf{a}}{k}}{\Pochhammersym{\mathbf{b}}{k}}\frac{z^{k}}{k!} = \frac{\Pochhammersym{\mathbf{a}}{m}z^{m}}{\Pochhammersym{\mathbf{b}}{m}m!}\genhyperF{q+2}{p}@@{-m,1,1-m-\mathbf{b}}{1-m-\mathbf{a}}{\frac{(-1)^{p+q+1}}{z}}		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}, GenerateConditions->None] == 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	Failed [258 / 300] Result: .9712138726+.322304451e-1*I Test Values: {a = -3/2, b = -3/2, p = 1/2*3^(1/2)+1/2*I, q = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, m = 1} Result: 1.825190824+.5153748995*I Test Values: {a = -3/2, b = -3/2, p = 1/2*3^(1/2)+1/2*I, q = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, m = 2} ... skip entries to safe data	Failed [258 / 300] Result: Complex[0.9712138727144698, 0.03223044535232533] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[m, 1], Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.8251908240859445, 0.5153749002123968] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[m, 2], Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data