Generalized hypergeometric function 3F2: Series representations (formula 07.27.06.0005)

HypergeometricPFQ

$Mathematica Notation$

Hypergeometric Functions

HypergeometricPFQ[{a₁,a₂,a₃},{b₁,b₂},z]

Series representations

Generalized power series

Expansions at z==1

For the function itself

General case

http://functions.wolfram.com/07.27.06.0005.01

Input Form

HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3]}, {Subscript[b, 1], Subscript[b, 2]}, z] == ((Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]])/ Product[Gamma[Subscript[a, k]], {k, 1, 3}]) (Sum[Subscript[g, k][0] (1 - z)^k, {k, 0, Infinity}] + (1 - z)^Subscript[\[Psi], 2] Sum[Subscript[g, k][Subscript[\[Psi], 2]] (1 - z)^k, {k, 0, Infinity}]) /; Abs[1 - z] < 1 && Subscript[g, 0][Subscript[\[Psi], 2]] == Gamma[-Subscript[\[Psi], 2]] && Subscript[g, k][r] == ((-1)^k/k!) Gamma[Subscript[a, 1] + r + k] Gamma[Subscript[a, 2] + r + k] Gamma[Subscript[\[Psi], 2] - 2 r - k] HypergeometricPFQRegularized[{Subscript[b, 1] - Subscript[a, 3], Subscript[b, 2] - Subscript[a, 3], Subscript[\[Psi], 2] - r - k}, {Subscript[a, 1] + Subscript[\[Psi], 2], Subscript[a, 2] + Subscript[\[Psi], 2]}, 1] && Subscript[\[Psi], 2] == Subscript[b, 1] + Subscript[b, 2] - Subscript[a, 1] - Subscript[a, 2] - Subscript[a, 3] && !Element[Subscript[\[Psi], 2], Integers] && Re[Subscript[a, 3]] > 0

Standard Form

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MathML Form

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type='integer'> 2 </cn> </apply> <integers /> </apply> <apply> <gt /> <apply> <real /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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},z]
HypergeometricPFQ[{a₁,a₂},{b₁,b₂,b₃},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]