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

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=-3/8

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

Input Form

Hypergeometric2F1[-(31/8), -(3/8), 6, z] == (524288 2^(1/4) (2 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (23363584 - 270962816 z + 1513455867 z^2 - 5717465005 z^3 + 19849456550 z^4 + 83118897246 z^5 + 1885948911 z^6 - 257127057 z^7 + 19411920 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 - 270962816 z + 1513455867 z^2 - 5717465005 z^3 + 19849456550 z^4 + 83118897246 z^5 + 1885948911 z^6 - 257127057 z^7 + 19411920 z^8) EllipticK[ 1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - Sqrt[1 - z] (23363584 - 270962816 z + 1513455867 z^2 - 5717465005 z^3 + 19849456550 z^4 + 83118897246 z^5 + 1885948911 z^6 - 257127057 z^7 + 19411920 z^8) EllipticK[1/2 - (1 - z)^(1/4)/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - (23363584 - 279724160 z + 1612671243 z^2 - 6258499387 z^3 + 21852245030 z^4 - 155288695254 z^5 - 69031618425 z^6 + 8028365697 z^7 - 1065714408 z^8 + 77647680 z^9) EllipticK[ 1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/ (60680285980058745 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]