Generalized hypergeometric function 5F4: Series representations (formula 07.29.06.0001)

HypergeometricPFQ

$Mathematica Notation$

Hypergeometric Functions

HypergeometricPFQ[{a₁,a₂,a₃,a₄,a₅},{b₁,b₂,b₃,b₄},z]

Series representations

Generalized power series

Expansions at z==1

http://functions.wolfram.com/07.29.06.0001.01

Input Form

HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3], Subscript[a, 4], Subscript[a, 5]}, {Subscript[b, 1], Subscript[b, 2], Subscript[b, 3], Subscript[b, 4]}, z] == (Product[Gamma[Subscript[b, k]], {k, 1, 4}]/Product[Gamma[Subscript[a, k]], {k, 1, 5}]) (Sum[Subscript[g, k][0] (1 - z)^k, {k, 0, Infinity}] + (1 - z)^Subscript[\[Psi], 4] Sum[Subscript[g, k][Subscript[\[Psi], 4]] (1 - z)^k, {k, 0, Infinity}]) /; Subscript[g, k][r] == ((-1)^k/k!) Gamma[Subscript[a, 1] + r + k] Gamma[Subscript[a, 2] + r + k] Gamma[Subscript[\[Psi], 4] - 2 r - k] Sum[((Pochhammer[Subscript[\[Psi], 4] - r - k, j] Pochhammer[Subscript[b, 2] + Subscript[b, 3] + Subscript[b, 4] - Subscript[a, 3] - Subscript[a, 4] - Subscript[a, 5], j] Pochhammer[Subscript[b, 1] - Subscript[a, 3], j])/ (Gamma[Subscript[\[Psi], 4] + Subscript[a, 1] + j] Gamma[Subscript[\[Psi], 4] + Subscript[a, 2] + j] j!)) Sum[((Pochhammer[Subscript[b, 4] + Subscript[b, 3] - Subscript[a, 5] - Subscript[a, 4], s] Pochhammer[Subscript[b, 2] - Subscript[a, 4], s] Pochhammer[-j, s])/(Pochhammer[Subscript[b, 2] + Subscript[b, 3] + Subscript[b, 4] - Subscript[a, 3] - Subscript[a, 4] - Subscript[a, 5], s] Pochhammer[ 1 - Subscript[b, 1] + Subscript[a, 3] - j, s] s!)) HypergeometricPFQ[{Subscript[b, 4] - Subscript[a, 5], Subscript[b, 3] - Subscript[a, 5], -s}, {Subscript[b, 4] + Subscript[b, 3] - Subscript[a, 5] - Subscript[a, 4], 1 - Subscript[b, 2] + Subscript[a, 4] - s}, 1], {s, 0, j}], {j, 0, Infinity}] && Subscript[\[Psi], 4] == Subscript[b, 1] + Subscript[b, 2] + Subscript[b, 3] + Subscript[b, 4] - Subscript[a, 1] - Subscript[a, 2] - Subscript[a, 3] - Subscript[a, 4] - Subscript[a, 5] && !Element[Subscript[\[Psi], 4], Integers]

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},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,a₆},{b₁,b₂,b₃,b₄,b₅},z]
HypergeometricPFQ[{a₁,...,a_p},{b₁,...,b_q},z]