Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.9045)

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Specific values

For rational parameters with denominators 4 and fixed z

For fixed z and a=-23/4, b>=a

For fixed z and a=-23/4, b=-19/4

http://functions.wolfram.com/07.23.03.9045.01

Input Form

Hypergeometric2F1[-(23/4), -(19/4), 11/2, z] == (32 (-2 (1074076080 - 28961694300 z + 440509927627 z^2 - 5848860835048 z^3 + 133018898423796 z^4 + 1622946543547120 z^5 + 3785631560463650 z^6 + 2776454582964744 z^7 + 654334135003204 z^8 + 37972691353292 z^9 + 78058255275 z^10) EllipticE[(1/2) (1 - Sqrt[z])] + 2 (1074076080 - 28961694300 z + 440509927627 z^2 - 5848860835048 z^3 + 133018898423796 z^4 + 1622946543547120 z^5 + 3785631560463650 z^6 + 2776454582964744 z^7 + 654334135003204 z^8 + 37972691353292 z^9 + 78058255275 z^10) EllipticE[(1/2) (1 + Sqrt[z])] + (1074076080 + 537038040 Sqrt[z] - 28961694300 z - 14436093980 z^(3/2) + 440509927627 z^2 + 219070603136 z^(5/2) - 5848860835048 z^3 - 2906664687696 z^(7/2) + 133018898423796 z^4 + 363469764800560 z^(9/2) + 1622946543547120 z^5 + 2255594602234520 z^(11/2) + 3785631560463650 z^6 + 3786140037216960 z^(13/2) + 2776454582964744 z^7 + 2173753258453552 z^(15/2) + 654334135003204 z^8 + 410322313675448 z^(17/2) + 37972691353292 z^9 + 18421748244900 z^(19/2) + 78058255275 z^10) EllipticK[(1/2) (1 - Sqrt[z])] - (1074076080 - 537038040 Sqrt[z] - 28961694300 z + 14436093980 z^(3/2) + 440509927627 z^2 - 219070603136 z^(5/2) - 5848860835048 z^3 + 2906664687696 z^(7/2) + 133018898423796 z^4 - 363469764800560 z^(9/2) + 1622946543547120 z^5 - 2255594602234520 z^(11/2) + 3785631560463650 z^6 - 3786140037216960 z^(13/2) + 2776454582964744 z^7 - 2173753258453552 z^(15/2) + 654334135003204 z^8 - 410322313675448 z^(17/2) + 37972691353292 z^9 - 18421748244900 z^(19/2) + 78058255275 z^10) EllipticK[(1/2) (1 + Sqrt[z])]) Gamma[3/4]^2)/ (9510247094902125 Pi^(3/2) z^(9/2))

Standard Form

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MathML Form

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type='integer'> 14436093980 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 28961694300 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 537038040 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> 1074076080 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 78058255275 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 10 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 18421748244900 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 19 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 37972691353292 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 9 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 410322313675448 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 17 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 654334135003204 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2173753258453552 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 15 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2776454582964744 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> 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type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2906664687696 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 7 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 5848860835048 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 219070603136 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 5 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 440509927627 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 14436093980 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 28961694300 </cn> <ci> z </ci> </apply> </apply> <apply> 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Rule Form

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Date Added to functions.wolfram.com (modification date)

2007-05-02

HypergeometricPFQ[{},{},z]
HypergeometricPFQ[{},{b},z]
HypergeometricPFQ[{a},{},z]
HypergeometricPFQ[{a},{b},z]
HypergeometricPFQ[{a₁},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂},{b₁,b₂,b₃},z]
HypergeometricPFQ[{a₁,a₂,a₃},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄},{b₁,b₂,b₃},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄,a₅},{b₁,b₂,b₃,b₄},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄,a₅,a₆},{b₁,b₂,b₃,b₄,b₅},z]
HypergeometricPFQ[{a₁,...,a_p},{b₁,...,b_q},z]