Gauss hypergeometric function 2F1: Representations through more general functions (formula 07.23.26.0206)

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/07.23.26.0206.01

Input Form

(Sqrt[z] + Sqrt[1 + z])^(1/2 + 2 a - c) Hypergeometric2F1[a, 1/2 + a, c, (Sqrt[z] - Sqrt[1 + z])/(2 Sqrt[z])] Hypergeometric2F1Regularized[ -(1/2) - a + c, -a + c, 1/2 - 2 a + c, (Sqrt[z] - Sqrt[1 + z])/ (2 Sqrt[z])] == ((2^(-3 - 2 a + 3 c) Gamma[c])/ (Pi Gamma[-1 - 2 a + 2 c])) MeijerG[{{(5 - 2 c)/4 + a}, {(1 + 2 c)/4 + a, (6 c - 1)/4 - a, (3 + 2 c)/4 - a}}, {{(1/4) (-1 + 2 c), c/2, (1/4) (1 + 2 c), (1 + c)/2}, {}}, z] /; !IntervalMemberQ[Interval[{-1, 0}], 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]