Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.bft2)

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Specific values

For rational parameters with denominators 8 and fixed z and a<0

For fixed z and a=-31/8, b>=a

For fixed z and a=-31/8, b=45/8

http://functions.wolfram.com/07.23.03.bft2.01

Input Form

Hypergeometric2F1[-(31/8), 45/8, 6, z] == (524288 2^(1/4) (-2 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (23363584 + 40064896 z + 84672315 z^2 + 241115210 z^3 + 1197665315 z^4 - 14980043232 z^5 + 31549469952 z^6 - 25347379200 z^7 + 7169310720 z^8) EllipticE[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (23363584 + 40064896 z + 84672315 z^2 + 241115210 z^3 + 1197665315 z^4 - 14980043232 z^5 + 31549469952 z^6 - 25347379200 z^7 + 7169310720 z^8) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + Sqrt[1 - z] (23363584 + 40064896 z + 84672315 z^2 + 241115210 z^3 + 1197665315 z^4 - 14980043232 z^5 + 31549469952 z^6 - 25347379200 z^7 + 7169310720 z^8) EllipticK[1/2 - (1 - z)^(1/4)/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + (23363584 + 31303552 z + 67252299 z^2 + 203982170 z^3 + 1095549455 z^4 + 24459737148 z^5 - 111115124736 z^6 + 171730910208 z^7 - 115130695680 z^8 + 28677242880 z^9) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/ (10464860466795735 Pi (1 + Sqrt[1 - z])^(1/4) z^5)

Standard Form

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MathML Form

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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]