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

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=37/8

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

Input Form

Hypergeometric2F1[-(31/8), 37/8, 6, z] == (524288 2^(1/4) (-2 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (-23363584 + 11773056 z + 17383653 z^2 + 35286370 z^3 + 127644825 z^4 - 1115242128 z^5 + 1899123072 z^6 - 1288894464 z^7 + 316293120 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 + 11773056 z + 17383653 z^2 + 35286370 z^3 + 127644825 z^4 - 1115242128 z^5 + 1899123072 z^6 - 1288894464 z^7 + 316293120 z^8) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + Sqrt[1 - z] (-23363584 + 11773056 z + 17383653 z^2 + 35286370 z^3 + 127644825 z^4 - 1115242128 z^5 + 1899123072 z^6 - 1288894464 z^7 + 316293120 z^8) EllipticK[1/2 - (1 - z)^(1/4)/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + (-23363584 + 20534400 z + 15364437 z^2 + 28833007 z^3 + 112821555 z^4 + 2069290377 z^5 - 7619193120 z^6 + 9932338944 z^7 - 5761806336 z^8 + 1265172480 z^9) EllipticK[1/2 - (1 - z)^(1/4)/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/(848502200010465 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]