Gauss hypergeometric function 2F1: Representations through more general functions (formula 07.23.26.0196)

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Representations through more general functions

Through Meijer G

Classical cases involving algebraic functions and ₂F^~₁with algebraic arguments

http://functions.wolfram.com/07.23.26.0196.01

Input Form

(1 + Sqrt[1 + z])^(1 + a + b - 2 c) Hypergeometric2F1[a, b, c, (1 - Sqrt[1 + z])/2] Hypergeometric2F1Regularized[1 - a, 1 - b, 1 - a - b + c, (1 - Sqrt[1 + z])/2] == (Gamma[c]/(Sqrt[Pi] Gamma[-a + c] Gamma[-b + c])) MeijerG[{{1 + a - c, 1 + b - c, (1 + a + b)/2 - c, (a + b)/2 + 1 - c}, {}}, {{0}, {1 + a + b - 2 c, 1 - c, a + b - c}}, z]

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)

2001-10-29

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]