Generalized hypergeometric function: Differential equations (formula 07.31.13.0009)

HypergeometricPFQ

$Mathematica Notation$

Hypergeometric Functions

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

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

Representation of fundamental system solutions near point z=ComplexInfinity for p>=q+1 in the general case

http://functions.wolfram.com/07.31.13.0009.01

Input Form

D[Fold[Function[{f, k}, z D[f, z] + (Subscript[b, k] - 1) f], w[z], {1, \[Ellipsis], q}], z] - Fold[Function[{f, l}, z D[f, z] + Subscript[a, l] f], w[z], {1, \[Ellipsis], p}] == 0 /; w[z] == Sum[(Subscript[c, k] HypergeometricPFQRegularized[ {Subscript[a, k], 1 + Subscript[a, k] - Subscript[b, 1], \[Ellipsis], 1 + Subscript[a, k] - Subscript[b, q]}, {1 + Subscript[a, k] - Subscript[a, 1], \[Ellipsis], 1 + Subscript[a, k] - Subscript[a, k - 1], 1 + Subscript[a, k] - Subscript[a, k + 1], \[Ellipsis], 1 + Subscript[a, k] - Subscript[a, p]}, (-1)^(1 - p + q)/z])/z^Subscript[a, k], {k, 1, p}] && ForAll[{j, k}, Element[{j, k}, Integers] && j != k && 1 <= j <= p && 1 <= k <= p, !Element[Subscript[a, j] - Subscript[a, k], 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)

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]