Regularized hypergeometric function 2F1: Series representations (formula 07.24.06.0040)

Hypergeometric2F1Regularized

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1Regularized[a,b,c,z]

Series representations

Generalized power series

Expansions at generic point z==z₀

For the function itself

http://functions.wolfram.com/07.24.06.0040.01

Input Form

Hypergeometric2F1Regularized[a, b, c, z] == (Pi/(Gamma[a] Gamma[b] Gamma[c - a] Gamma[c - b])) Sum[((-1)^k/k!) (Gamma[a + k] Gamma[b + k] (-2 I E^(I (c - a - b) Pi Floor[Arg[-z + Subscript[z, 0]]/(2 Pi)]) Floor[(Pi + Arg[1 - Subscript[z, 0]])/(2 Pi)] Floor[Arg[-z + Subscript[z, 0]]/(2 Pi)] + Csc[(c - a - b) Pi] (1/(1 - Subscript[z, 0]))^((c - a - b) Floor[Arg[-z + Subscript[z, 0]]/(2 Pi)]) (1 - Subscript[z, 0])^ ((c - a - b) Floor[Arg[-z + Subscript[z, 0]]/(2 Pi)])) Hypergeometric2F1Regularized[a + k, b + k, 1 + a + b - c + k, 1 - Subscript[z, 0]] - Gamma[-a + c] Gamma[-b + c] (1 - Subscript[z, 0])^(-a - b + c - k) Csc[(c - a - b) Pi] (1/(1 - Subscript[z, 0]))^((c - a - b) Floor[Arg[-z + Subscript[z, 0]]/ (2 Pi)]) (1 - Subscript[z, 0])^((c - a - b) Floor[Arg[-z + Subscript[z, 0]]/(2 Pi)]) Hypergeometric2F1Regularized[-a + c, -b + c, 1 - a - b + c - k, 1 - Subscript[z, 0]]) (z - Subscript[z, 0])^k, {k, 0, Infinity}] /; !Element[c - a - b, Integers]

Standard Form

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MathML Form

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Date Added to functions.wolfram.com (modification date)

2007-05-02

HypergeometricPFQRegularized[{},{b},z]
HypergeometricPFQRegularized[{a₁},{b₁},z]
HypergeometricPFQRegularized[{a₁,...,a_p},{b₁,...,b_q},z]