Regularized hypergeometric function 2F1: Representations through more general functions (formula 07.24.26.0260)

Hypergeometric2F1Regularized

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1Regularized[a,b,c,z]

Representations through more general functions

Through Meijer G

Generalized cases involving algebraic functions and ₂F₁with algebraic arguments

http://functions.wolfram.com/07.24.26.0260.01

Input Form

(z + Sqrt[1 + z^2])^(1 + a + b - 2 c) Hypergeometric2F1Regularized[a, b, c, (z - Sqrt[1 + z^2])/(2 z)] Hypergeometric2F1[1 - a, 1 - b, 1 - a - b + c, (z - Sqrt[1 + z^2])/(2 z)] == (Gamma[1 - a - b + c]/(Sqrt[Pi] Gamma[-a + c] Gamma[-b + c])) MeijerG[{{(3 + a + b)/2 - c}, {(1 - a - b)/2 + c, (1 + a + b)/2, (3 - a - b)/2}}, {{1/2, 1, (1 - a + b)/2, (1 + a - b)/2}, {}}, z, 1/2] /; Re[z] > 0

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]