Generalized hypergeometric function 2F2: Series representations (formula 07.25.06.0011)

HypergeometricPFQ

$Mathematica Notation$

Hypergeometric Functions

HypergeometricPFQ[{a₁,a₂},{b₁,b₂},z]

Series representations

Asymptotic series expansions

Case of double poles

http://functions.wolfram.com/07.25.06.0011.01

Input Form

HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 1]}, {Subscript[b, 1], Subscript[b, 2]}, z] \[Proportional] ((Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]])/Gamma[Subscript[a, 1]]^2) z^\[Chi] E^z (1 + O[1/z]) + (((Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]])/ (Gamma[Subscript[a, 1]] Gamma[Subscript[b, 1] - Subscript[a, 1]] Gamma[Subscript[b, 2] - Subscript[a, 1]])) (Log[-z] (1 + O[1/z]) - (2 EulerGamma + PolyGamma[Subscript[a, 1]] + PolyGamma[Subscript[b, 1] - Subscript[a, 1]] + PolyGamma[Subscript[b, 2] - Subscript[a, 1]]) (1 + O[1/z])))/ (-z)^Subscript[a, 1] /; (Abs[z] -> Infinity) && \[Chi] == 2 Subscript[a, 1] - Subscript[b, 1] - Subscript[b, 2]

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},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]