Hypergeometric Function - 15.2 Definitions and Analytical Properties
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| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
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| 15.2.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \hyperF@{a}{b}{c}{z} = \sum_{s=0}^{\infty}\frac{\Pochhammersym{a}{s}\Pochhammersym{b}{s}}{\Pochhammersym{c}{s}s!}z^{s}}
\hyperF@{a}{b}{c}{z} = \sum_{s=0}^{\infty}\frac{\Pochhammersym{a}{s}\Pochhammersym{b}{s}}{\Pochhammersym{c}{s}s!}z^{s} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | hypergeom([a, b], [c], z) = sum((pochhammer(a, s)*pochhammer(b, s))/(pochhammer(c, s)*factorial(s))*(z)^(s), s = 0..infinity)
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Hypergeometric2F1[a, b, c, z] == Sum[Divide[Pochhammer[a, s]*Pochhammer[b, s],Pochhammer[c, s]*(s)!]*(z)^(s), {s, 0, Infinity}, GenerateConditions->None]
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Failure | Successful | Skipped - Because timed out | Successful [Tested: 300] |
| 15.2.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \hyperOlverF@{a}{b}{c}{z} = \sum_{s=0}^{\infty}\frac{\Pochhammersym{a}{s}\Pochhammersym{b}{s}}{\EulerGamma@{c+s}s!}z^{s}}
\hyperOlverF@{a}{b}{c}{z} = \sum_{s=0}^{\infty}\frac{\Pochhammersym{a}{s}\Pochhammersym{b}{s}}{\EulerGamma@{c+s}s!}z^{s} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |z| < 1, \realpart@@{(c+s)} > 0} | hypergeom([a, b], [c], z)/GAMMA(c) = sum((pochhammer(a, s)*pochhammer(b, s))/(GAMMA(c + s)*factorial(s))*(z)^(s), s = 0..infinity)
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Hypergeometric2F1Regularized[a, b, c, z] == Sum[Divide[Pochhammer[a, s]*Pochhammer[b, s],Gamma[c + s]*(s)!]*(z)^(s), {s, 0, Infinity}, GenerateConditions->None]
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Successful | Successful | - | Failed [25 / 216]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, 0.5]}
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, 1.5], Rule[c, -2], Rule[z, 0.5]}
... skip entries to safe data |
| 15.2.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \hyperOlverF@@{a}{b}{c}{x+\iunit 0}-\hyperOlverF@@{a}{b}{c}{x-\iunit 0} = \frac{2\pi\iunit}{\EulerGamma@{a}\EulerGamma@{b}}(x-1)^{c-a-b}\hyperOlverF@@{c-a}{c-b}{c-a-b+1}{1-x}}
\hyperOlverF@@{a}{b}{c}{x+\iunit 0}-\hyperOlverF@@{a}{b}{c}{x-\iunit 0} = \frac{2\pi\iunit}{\EulerGamma@{a}\EulerGamma@{b}}(x-1)^{c-a-b}\hyperOlverF@@{c-a}{c-b}{c-a-b+1}{1-x} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle x > 1, \realpart@@{a} > 0, \realpart@@{b} > 0, |(x+\iunit 0)| < 1, |(x-\iunit 0)| < 1, |(1-x)| < 1, \realpart@@{(c+s)} > 0, \realpart@@{((c-a-b+1)+s)} > 0} | 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)
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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]
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Failure | Failure | Error | Skip - No test values generated |
| 15.2.E3_5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lim_{c\to-n}\frac{\hyperF@{a}{b}{c}{z}}{\EulerGamma@{c}} = \hyperOlverF@{a}{b}{-n}{z}}
\lim_{c\to-n}\frac{\hyperF@{a}{b}{c}{z}}{\EulerGamma@{c}} = \hyperOlverF@{a}{b}{-n}{z} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{c} > 0, |z| < 1, \realpart@@{((-n)+s)} > 0} | limit((hypergeom([a, b], [c], z))/(GAMMA(c)), c = - n) = hypergeom([a, b], [- n], z)/GAMMA(- n)
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Limit[Divide[Hypergeometric2F1[a, b, c, z],Gamma[c]], c -> - n, GenerateConditions->None] == Hypergeometric2F1Regularized[a, b, - n, z]
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Failure | Successful | Successful [Tested: 0] | Failed [25 / 36]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 3], Rule[z, 0.5]}
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, 1.5], Rule[n, 3], Rule[z, 0.5]}
... skip entries to safe data |
| 15.2.E3_5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \hyperOlverF@{a}{b}{-n}{z} = \frac{\Pochhammersym{a}{n+1}\Pochhammersym{b}{n+1}}{(n+1)!}z^{n+1}\hyperF@{a+n+1}{b+n+1}{n+2}{z}}
\hyperOlverF@{a}{b}{-n}{z} = \frac{\Pochhammersym{a}{n+1}\Pochhammersym{b}{n+1}}{(n+1)!}z^{n+1}\hyperF@{a+n+1}{b+n+1}{n+2}{z} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{c} > 0, |z| < 1, \realpart@@{((-n)+s)} > 0} | 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)
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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]
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Failure | Failure | Failed [25 / 36] Result: Float(undefined)+Float(undefined)*I
Test Values: {a = -3/2, b = -3/2, z = 1/2, n = 3}
Result: Float(undefined)+Float(undefined)*I
Test Values: {a = -3/2, b = 3/2, z = 1/2, n = 3}
... skip entries to safe data |
Successful [Tested: 180] |
| 15.2.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \hyperF@{-m}{b}{c}{z} = \sum_{n=0}^{m}\frac{\Pochhammersym{-m}{n}\Pochhammersym{b}{n}}{\Pochhammersym{c}{n}{n!}}z^{n}}
\hyperF@{-m}{b}{c}{z} = \sum_{n=0}^{m}\frac{\Pochhammersym{-m}{n}\Pochhammersym{b}{n}}{\Pochhammersym{c}{n}{n!}}z^{n} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | hypergeom([- m, b], [c], z) = sum((pochhammer(- m, n)*pochhammer(b, n))/(pochhammer(c, n)*factorial(n))*(z)^(n), n = 0..m)
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Hypergeometric2F1[- m, b, c, z] == Sum[Divide[Pochhammer[- m, n]*Pochhammer[b, n],Pochhammer[c, n]*(n)!]*(z)^(n), {n, 0, m}, GenerateConditions->None]
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Successful | Successful | - | Successful [Tested: 300] |
| 15.2.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{n=0}^{m}\frac{\Pochhammersym{-m}{n}\Pochhammersym{b}{n}}{\Pochhammersym{c}{n}{n!}}z^{n} = \sum_{n=0}^{m}(-1)^{n}\binom{m}{n}\frac{\Pochhammersym{b}{n}}{\Pochhammersym{c}{n}}z^{n}}
\sum_{n=0}^{m}\frac{\Pochhammersym{-m}{n}\Pochhammersym{b}{n}}{\Pochhammersym{c}{n}{n!}}z^{n} = \sum_{n=0}^{m}(-1)^{n}\binom{m}{n}\frac{\Pochhammersym{b}{n}}{\Pochhammersym{c}{n}}z^{n} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | 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)
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Sum[Divide[Pochhammer[- m, n]*Pochhammer[b, n],Pochhammer[c, n]*(n)!]*(z)^(n), {n, 0, m}, GenerateConditions->None] == Sum[(- 1)^(n)*Binomial[m,n]*Divide[Pochhammer[b, n],Pochhammer[c, n]]*(z)^(n), {n, 0, m}, GenerateConditions->None]
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Successful | Successful | - | Successful [Tested: 300] |
| 15.2.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \hyperF@@{-m}{b}{-m-\ell}{z} = \lim_{c\to-m-\ell}\left(\lim_{a\to-m}\hyperF@@{a}{b}{c}{z}\right)}
\hyperF@@{-m}{b}{-m-\ell}{z} = \lim_{c\to-m-\ell}\left(\lim_{a\to-m}\hyperF@@{a}{b}{c}{z}\right) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | hypergeom([- m, b], [- m - ell], z) = limit(limit(hypergeom([a, b], [c], z), a = - m), c = - m - ell)
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Hypergeometric2F1[- m, b, - m - \[ScriptL], z] == Limit[Limit[Hypergeometric2F1[a, b, c, z], a -> - m, GenerateConditions->None], c -> - m - \[ScriptL], GenerateConditions->None]
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Failure | Successful | Successful [Tested: 126] | Successful [Tested: 126] |
| 15.2.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \hyperF@@{-m}{b}{-m-\ell}{z} = \lim_{a\to-m}\hyperF@@{a}{b}{a-\ell}{z}}
\hyperF@@{-m}{b}{-m-\ell}{z} = \lim_{a\to-m}\hyperF@@{a}{b}{a-\ell}{z} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | hypergeom([- m, b], [- m - ell], z) = limit(hypergeom([a, b], [a - ell], z), a = - m)
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Hypergeometric2F1[- m, b, - m - \[ScriptL], z] == Limit[Hypergeometric2F1[a, b, a - \[ScriptL], z], a -> - m, GenerateConditions->None]
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Failure | Successful | Successful [Tested: 0] | Successful [Tested: 126] |