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

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=-33/8, b>=a

For fixed z and a=-33/8, b=-27/8

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

Input Form

Hypergeometric2F1[-(33/8), -(27/8), 6, z] == (262144 2^(3/4) (1 + Sqrt[1 - z])^(1/4) (4 (48660480 - 927400320 z + 9242010855 z^2 - 69869275245 z^3 + 575368748955 z^4 + 30507765220807 z^5 + 60846689148357 z^6 + 28146952595817 z^7 + 2866083076857 z^8 + 20143021437 z^9) EllipticE[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 3 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] Sqrt[1 - z] (-6082560 + 111505680 z - 1074664305 z^2 + 7960499910 z^3 + 6851738171025 z^4 + 15325445656996 z^5 + 7689417932097 z^6 + 845066592678 z^7 + 6714340479 z^8) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 20 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1520640 - 28731780 z + 284175045 z^2 - 2138189130 z^3 + 1055335088115 z^4 + 2796444383936 z^5 + 1896930298683 z^6 + 380694928878 z^7 + 17551316013 z^8) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 2 (48660480 - 927400320 z + 9242010855 z^2 - 69869275245 z^3 + 575368748955 z^4 + 30507765220807 z^5 + 60846689148357 z^6 + 28146952595817 z^7 + 2866083076857 z^8 + 20143021437 z^9) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/ (5440495299748584075 Pi 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]