Generalized hypergeometric function: Series representations (formula 07.31.06.0047)

HypergeometricPFQ

$Mathematica Notation$

Hypergeometric Functions

HypergeometricPFQ[{a₁,...,a_p},{b₁,...,b_q},z]

Series representations

Generalized power series

Expansions on branch cuts for p==q+1

For the function itself

http://functions.wolfram.com/07.31.06.0047.01

Input Form

HypergeometricPFQ[{Subscript[a, 1], \[Ellipsis], Subscript[a, q + 1]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, z] == (Product[Gamma[Subscript[b, k]], {k, 1, q}]/Product[Gamma[Subscript[a, k]], {k, 1, q + 1}]) Sum[(1/k!) MeijerG[{{1 - Subscript[a, 1] - k, \[Ellipsis], 1 - Subscript[a, q + 1] - k}, {}}, {{0}, {1 - Subscript[b, 1] - k, \[Ellipsis], 1 - Subscript[b, q] - k}}, (-x) E^(2 Pi I Floor[Arg[x - z]/(2 Pi)])] (z - x)^k, {k, 0, Infinity}] /; x > 1

Standard Form

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MathML Form

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

2007-05-02

HypergeometricPFQ[{},{},z]
HypergeometricPFQ[{},{b},z]
HypergeometricPFQ[{a},{},z]
HypergeometricPFQ[{a},{b},z]
HypergeometricPFQ[{a₁},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂},{b₁},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]