Regularized hypergeometric function 2F1: Series representations (formula 07.24.06.0043)

Hypergeometric2F1Regularized

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1Regularized[a,b,c,z]

Series representations

Generalized power series

Expansions on branch cuts

For the function itself

http://functions.wolfram.com/07.24.06.0043.01

Input Form

Hypergeometric2F1Regularized[a, b, c, z] \[Proportional] (1/(Gamma[a] Gamma[b] Gamma[-a + c] Gamma[-b + c])) (E^(I (c - a - b) Pi Floor[Arg[x - z]/(2 Pi)]) (-2 I Pi Floor[Arg[x - z]/(2 Pi)] Gamma[a] Gamma[b] Hypergeometric2F1Regularized[a, b, 1 + a + b - c, 1 - x] + E^(I (c - a - b) Pi Floor[Arg[x - z]/(2 Pi)]) MeijerG[{{1 - a, 1 - b}, {}}, {{0, -a - b + c}, {}}, 1 - x]) + E^(I (c - a - b) Pi Floor[Arg[x - z]/(2 Pi)]) (2 I Pi Floor[Arg[x - z]/(2 Pi)] Gamma[1 + a] Gamma[1 + b] Hypergeometric2F1Regularized[1 + a, 1 + b, 2 + a + b - c, 1 - x] + E^(I (c - a - b) Pi Floor[Arg[x - z]/(2 Pi)]) MeijerG[{{-a, -b}, {}}, {{0, -1 - a - b + c}, {}}, 1 - x]) (z - x) + (1/2) E^(I (c - a - b) Pi Floor[Arg[x - z]/(2 Pi)]) (-2 I Pi Floor[Arg[x - z]/(2 Pi)] Gamma[2 + a] Gamma[2 + b] Hypergeometric2F1Regularized[2 + a, 2 + b, 3 + a + b - c, 1 - x] + E^(I (c - a - b) Pi Floor[Arg[x - z]/(2 Pi)]) MeijerG[{{-1 - a, -1 - b}, {}}, {{0, -2 - a - b + c}, {}}, 1 - x]) (z - x)^2 + O[(z - x)^3]) /; Element[x, Reals] && x > 1

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)

2007-05-02

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