Regularized hypergeometric function 2F1: Series representations (formula 07.24.06.0045)

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/07.24.06.0045.01

Input Form

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