Regularized hypergeometric function 2F1: Representations through more general functions (formula 07.24.26.0182)

Hypergeometric2F1Regularized

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1Regularized[a,b,c,z]

Representations through more general functions

Through Meijer G

Classical cases involving algebraic functions and ₂F₁with algebraic arguments

http://functions.wolfram.com/07.24.26.0182.01

Input Form

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

HypergeometricPFQRegularized[{},{b},z]
HypergeometricPFQRegularized[{a₁},{b₁},z]
HypergeometricPFQRegularized[{a₁,...,a_p},{b₁,...,b_q},z]