Gauss hypergeometric function 2F1: Representations through more general functions (formula 07.23.26.0231)

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Representations through more general functions

Through Meijer G

Generalized cases for products of ₂F₁ with algebraic arguments

http://functions.wolfram.com/07.23.26.0231.01

Input Form

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

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]