Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.ag69)

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=-5/4, b>=a

For fixed z and a=-5/4, b=15/4

http://functions.wolfram.com/07.23.03.ag69.01

Input Form

Hypergeometric2F1[-(5/4), 15/4, -(11/2), z] == (1/(13660416 Pi^(3/2))) (((1/(-1 + z)^8) (2 (3415104 - 23983344 z + 69893208 z^2 - 105327915 z^3 + 77917532 z^4 - 9291282 z^5 - 15723708 z^6 + 19297941 z^7 - 8380320 z^8 + 1357824 z^9) EllipticE[(1/2) (1 - Sqrt[z])]) + (1/(-1 + z)^8) (2 (3415104 - 23983344 z + 69893208 z^2 - 105327915 z^3 + 77917532 z^4 - 9291282 z^5 - 15723708 z^6 + 19297941 z^7 - 8380320 z^8 + 1357824 z^9) EllipticE[(1/2) (1 + Sqrt[z])]) + (1/((-1 + Sqrt[z])^8 (1 + Sqrt[z])^7)) ((-3415104 + 1707552 Sqrt[z] + 22275792 z - 10426416 z^(3/2) - 59466792 z^2 + 25448346 z^(5/2) + 79879569 z^3 - 29674337 z^(7/2) - 48243195 z^4 + 12507495 z^(9/2) - 3216213 z^5 + 6075069 z^(11/2) + 9648639 z^6 - 13935597 z^(13/2) - 5362344 z^7 + 7361952 z^(15/2) + 1018368 z^8 - 1357824 z^(17/2)) EllipticK[(1/2) (1 - Sqrt[z])]) + (1/((-1 + Sqrt[z])^7 (1 + Sqrt[z])^8)) ((3415104 + 1707552 Sqrt[z] - 22275792 z - 10426416 z^(3/2) + 59466792 z^2 + 25448346 z^(5/2) - 79879569 z^3 - 29674337 z^(7/2) + 48243195 z^4 + 12507495 z^(9/2) + 3216213 z^5 + 6075069 z^(11/2) - 9648639 z^6 - 13935597 z^(13/2) + 5362344 z^7 + 7361952 z^(15/2) - 1018368 z^8 - 1357824 z^(17/2)) EllipticK[(1/2) (1 + Sqrt[z])])) Gamma[1/4]^2)

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]