Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.aayx)

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=-17/4, b>=a

For fixed z and a=-17/4, b=23/4

http://functions.wolfram.com/07.23.03.aayx.01

Input Form

Hypergeometric2F1[-(17/4), 23/4, -(9/2), z] == (1/(176964480 Pi^(3/2))) (((1/(-1 + z)^6) (4 (22120560 - 9831360 z - 12618375 z^2 - 25440107 z^3 - 83491947 z^4 - 760565421 z^5 + 5493498970 z^6 - 11579827200 z^7 + 11432472576 z^8 - 5519704064 z^9 + 1056964608 z^10) EllipticE[(1/2) (1 - Sqrt[z])]) + (1/(-1 + z)^6) (4 (22120560 - 9831360 z - 12618375 z^2 - 25440107 z^3 - 83491947 z^4 - 760565421 z^5 + 5493498970 z^6 - 11579827200 z^7 + 11432472576 z^8 - 5519704064 z^9 + 1056964608 z^10) EllipticE[(1/2) (1 + Sqrt[z])]) + (1/((-1 + Sqrt[z])^6 (1 + Sqrt[z])^5)) ((-44241120 + 22120560 Sqrt[z] - 2457840 z + 10445820 z^(3/2) + 14790930 z^2 - 2274965 z^(5/2) + 53155179 z^3 - 26851902 z^(7/2) + 193835796 z^4 - 107868453 z^(9/2) + 1628999295 z^5 - 3378184820 z^(11/2) - 7608813120 z^6 + 12157992960 z^(13/2) + 11001661440 z^7 - 16022200320 z^(15/2) - 6842744832 z^8 + 9453961216 z^(17/2) + 1585446912 z^9 - 2113929216 z^(19/2)) EllipticK[(1/2) (1 - Sqrt[z])]) - (1/((-1 + Sqrt[z])^5 (1 + Sqrt[z])^6)) ((-44241120 - 22120560 Sqrt[z] - 2457840 z - 10445820 z^(3/2) + 14790930 z^2 + 2274965 z^(5/2) + 53155179 z^3 + 26851902 z^(7/2) + 193835796 z^4 + 107868453 z^(9/2) + 1628999295 z^5 + 3378184820 z^(11/2) - 7608813120 z^6 - 12157992960 z^(13/2) + 11001661440 z^7 + 16022200320 z^(15/2) - 6842744832 z^8 - 9453961216 z^(17/2) + 1585446912 z^9 + 2113929216 z^(19/2)) EllipticK[(1/2) (1 + Sqrt[z])])) Gamma[1/4]^2)

Standard Form

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MathML Form

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type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 107868453 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 9 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 193835796 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 26851902 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 7 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 53155179 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2274965 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 5 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 14790930 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 10445820 </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'> 2457840 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 22120560 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -44241120 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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]