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

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=-39/8, b>=a

For fixed z and a=-39/8, b=37/8

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

Input Form

Hypergeometric2F1[-(39/8), 37/8, 6, z] == (524288 2^(1/4) (2 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (23363584 - 13963392 z - 22152197 z^2 - 50034775 z^3 - 211044435 z^4 + 2445469103 z^5 - 5624402784 z^6 + 5764720896 z^7 - 2841907200 z^8 + 551485440 z^9) 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 - 13963392 z - 22152197 z^2 - 50034775 z^3 - 211044435 z^4 + 2445469103 z^5 - 5624402784 z^6 + 5764720896 z^7 - 2841907200 z^8 + 551485440 z^9) EllipticK[ 1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - Sqrt[1 - z] (23363584 - 13963392 z - 22152197 z^2 - 50034775 z^3 - 211044435 z^4 + 2445469103 z^5 - 5624402784 z^6 + 5764720896 z^7 - 2841907200 z^8 + 551485440 z^9) EllipticK[ 1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - (23363584 - 22724736 z - 19311605 z^2 - 41568613 z^3 - 190082235 z^4 - 4025711767 z^5 + 18634120428 z^6 - 32453752320 z^7 + 28271299584 z^8 - 12424642560 z^9 + 2205941760 z^10) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/ (1718760866687865 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]