Gauss hypergeometric function 2F1: Representations through more general functions (formula 07.23.26.0121)

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Representations through more general functions

Through Meijer G

Classical cases involving products of ₂F₁ with algebraic arguments

http://functions.wolfram.com/07.23.26.0121.01

Input Form

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

Standard Form

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MathML Form

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</mo> <mrow> <mi> z </mi> <mo> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>

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]