Hypergeometric Function - 15.16 Products

From testwiki
Jump to navigation Jump to search


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
15.16.E1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \hyperF@@{a}{b}{c-\frac{1}{2}}{z}\hyperF@@{c-a}{c-b}{c+\frac{1}{2}}{z} = \sum_{s=0}^{\infty}\frac{\Pochhammersym{c}{s}}{\Pochhammersym{c+\frac{1}{2}}{s}}A_{s}z^{s}}
\hyperF@@{a}{b}{c-\frac{1}{2}}{z}\hyperF@@{c-a}{c-b}{c+\frac{1}{2}}{z} = \sum_{s=0}^{\infty}\frac{\Pochhammersym{c}{s}}{\Pochhammersym{c+\frac{1}{2}}{s}}A_{s}z^{s}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |z| < 1} 
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}, GenerateConditions->None]
Failure Failure Skipped - Because timed out Skipped - Because timed out
15.16.E2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (1-z)^{a+b-c}\hyperF@{2a}{2b}{2c-1}{z} = \sum_{s=0}^{\infty}A_{s}z^{s}}
(1-z)^{a+b-c}\hyperF@{2a}{2b}{2c-1}{z} = \sum_{s=0}^{\infty}A_{s}z^{s}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |z| < 1} 
(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}, GenerateConditions->None]
Failure Failure
Failed [300 / 300]
Result: -1.113332374-1.*I
Test Values: {a = -3/2, b = -3/2, c = -3/2, z = 1/2, A[s] = 1/2*3^(1/2)+1/2*I}


Result: 1.618718434-1.732050808*I
Test Values: {a = -3/2, b = -3/2, c = -3/2, z = 1/2, A[s] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.20011980854170835, -0.8439394617218601]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[A, s], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[1.3278316576066613, -0.6694818315348507]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[A, s], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
15.16.E3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \hyperF@@{a}{b}{c}{z}\hyperF@@{a}{b}{c}{\zeta} = \sum_{s=0}^{\infty}\frac{\Pochhammersym{a}{s}\Pochhammersym{b}{s}\Pochhammersym{c-a}{s}\Pochhammersym{c-b}{s}}{\Pochhammersym{c}{s}\Pochhammersym{c}{2s}s!}\left(z\zeta\right)^{s}\hyperF@@{a+s}{b+s}{c+2s}{z+\zeta-z\zeta}}
\hyperF@@{a}{b}{c}{z}\hyperF@@{a}{b}{c}{\zeta} = \sum_{s=0}^{\infty}\frac{\Pochhammersym{a}{s}\Pochhammersym{b}{s}\Pochhammersym{c-a}{s}\Pochhammersym{c-b}{s}}{\Pochhammersym{c}{s}\Pochhammersym{c}{2s}s!}\left(z\zeta\right)^{s}\hyperF@@{a+s}{b+s}{c+2s}{z+\zeta-z\zeta}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |z| < 1, |\zeta| < 1, |z+\zeta-z\zeta| < 1} 
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}, GenerateConditions->None]
Failure Failure Skipped - Because timed out Skipped - Because timed out
15.16.E4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \hyperF@@{a}{b}{c}{z}\hyperF@@{-a}{-b}{-c}{z}+\frac{ab(a-c)(b-c)}{c^{2}(1-c^{2})}z^{2}\hyperF@@{1+a}{1+b}{2+c}{z}\hyperF@@{1-a}{1-b}{2-c}{z} = 1}
\hyperF@@{a}{b}{c}{z}\hyperF@@{-a}{-b}{-c}{z}+\frac{ab(a-c)(b-c)}{c^{2}(1-c^{2})}z^{2}\hyperF@@{1+a}{1+b}{2+c}{z}\hyperF@@{1-a}{1-b}{2-c}{z} = 1		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Skipped - Because timed out
Failed [98 / 300]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
15.16.E5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \hyperF@@{\frac{1}{2}+\lambda}{-\frac{1}{2}-\nu}{1+\lambda+\mu}{z}\hyperF@@{\frac{1}{2}-\lambda}{\frac{1}{2}+\nu}{1+\nu+\mu}{1-z}+\hyperF@@{\frac{1}{2}+\lambda}{\frac{1}{2}-\nu}{1+\lambda+\mu}{z}\hyperF@@{-\frac{1}{2}-\lambda}{\frac{1}{2}+\nu}{1+\nu+\mu}{1-z}-\hyperF@@{\frac{1}{2}+\lambda}{\frac{1}{2}-\nu}{1+\lambda+\mu}{z}\hyperF@@{\frac{1}{2}-\lambda}{\frac{1}{2}+\nu}{1+\nu+\mu}{1-z} = \frac{\EulerGamma@{1+\lambda+\mu}\EulerGamma@{1+\nu+\mu}}{\EulerGamma@{\lambda+\mu+\nu+\frac{3}{2}}\EulerGamma@{\frac{1}{2}+\nu}}}
\hyperF@@{\frac{1}{2}+\lambda}{-\frac{1}{2}-\nu}{1+\lambda+\mu}{z}\hyperF@@{\frac{1}{2}-\lambda}{\frac{1}{2}+\nu}{1+\nu+\mu}{1-z}+\hyperF@@{\frac{1}{2}+\lambda}{\frac{1}{2}-\nu}{1+\lambda+\mu}{z}\hyperF@@{-\frac{1}{2}-\lambda}{\frac{1}{2}+\nu}{1+\nu+\mu}{1-z}-\hyperF@@{\frac{1}{2}+\lambda}{\frac{1}{2}-\nu}{1+\lambda+\mu}{z}\hyperF@@{\frac{1}{2}-\lambda}{\frac{1}{2}+\nu}{1+\nu+\mu}{1-z} = \frac{\EulerGamma@{1+\lambda+\mu}\EulerGamma@{1+\nu+\mu}}{\EulerGamma@{\lambda+\mu+\nu+\frac{3}{2}}\EulerGamma@{\frac{1}{2}+\nu}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\phase@@{z}| < \pi, |\phase@{1-z}| < \pi, \realpart@@{(1+\lambda+\mu)} > 0, \realpart@@{(1+\nu+\mu)} > 0, \realpart@@{(\lambda+\mu+\nu+\frac{3}{2})} > 0, \realpart@@{(\frac{1}{2}+\nu)} > 0} 
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 Skipped - Because timed out
Failed [251 / 300]
Result: Complex[0.3564253165633178, -0.5060695815565636]
Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[λ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[μ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ν, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}


Result: Complex[0.011198613289511883, 0.30916385360889426]
Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[λ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[μ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ν, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]]}

... skip entries to safe data
Retrieved from "https://lct.wmflabs.org/w/index.php?title=15.16&oldid=58311"