Gauss hypergeometric function 2F1: Representations through more general functions (formula 07.23.26.0209)

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/07.23.26.0209.01

Input Form

(Sqrt[z] + Sqrt[1 + z])^((1/2) (-1 + a + b)) Hypergeometric2F1[a, b, (1/2) (1 + a + b), (Sqrt[z] - Sqrt[1 + z])/(2 Sqrt[z])] Hypergeometric2F1Regularized[(1/2) (1 - a + b), (1/2) (1 + a - b), (1/2) (3 - a - b), (Sqrt[z] - Sqrt[1 + z])/(2 Sqrt[z])] == (1/Pi^(3/2)) 2^((1/2) (-1 + a + b)) Cos[(1/2) (a - b) Pi] Gamma[(1/2) (1 + a + b)] MeijerG[{{(1/4) (3 + a + b)}, {(1/4) (1 + 3 a + 3 b), (1/4) (5 - a - b)}}, {{(1/4) (1 + a + b), (1/4) (1 + 3 a - b), (1/4) (1 - a + 3 b)}, {}}, 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]