Generalized hypergeometric function 2F3: Differential equations (formula 07.26.13.0004)

HypergeometricPFQ

$Mathematica Notation$

Hypergeometric Functions

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

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

http://functions.wolfram.com/07.26.13.0004.01

Input Form

z^3 Derivative[4][w][z] + (Subscript[b, 1] + Subscript[b, 2] + Subscript[b, 3] + 3) z^2 Derivative[3][w][z] + (Subscript[b, 1] Subscript[b, 2] + Subscript[b, 2] Subscript[b, 3] + Subscript[b, 1] Subscript[b, 3] + Subscript[b, 1] + Subscript[b, 2] + Subscript[b, 3] + 1 - z) z Derivative[2][w][z] + (Subscript[b, 1] Subscript[b, 2] Subscript[b, 3] - (Subscript[a, 1] + Subscript[a, 2] + 1) z) Derivative[1][w][z] - Subscript[a, 1] Subscript[a, 2] w[z] == 0 /; w[z] == Subscript[c, 1] HypergeometricPFQRegularized[ {Subscript[a, 1], Subscript[a, 2]}, {Subscript[b, 1], Subscript[b, 2], Subscript[b, 3]}, z] + Subscript[c, 2] z^(1 - Subscript[b, 1]) HypergeometricPFQRegularized[{1 + Subscript[a, 1] - Subscript[b, 1], 1 + Subscript[a, 2] - Subscript[b, 1]}, {2 - Subscript[b, 1], 1 - Subscript[b, 1] + Subscript[b, 2], 1 - Subscript[b, 1] + Subscript[b, 3]}, z] + Subscript[c, 3] z^(1 - Subscript[b, 2]) HypergeometricPFQRegularized[{1 + Subscript[a, 1] - Subscript[b, 2], 1 + Subscript[a, 2] - Subscript[b, 2]}, {2 - Subscript[b, 2], 1 + Subscript[b, 1] - Subscript[b, 2], 1 - Subscript[b, 2] + Subscript[b, 3]}, z] + Subscript[c, 4] z^(1 - Subscript[b, 3]) HypergeometricPFQRegularized[{1 + Subscript[a, 1] - Subscript[b, 3], 1 + Subscript[a, 2] - Subscript[b, 3]}, {2 - Subscript[b, 3], 1 + Subscript[b, 1] - Subscript[b, 3], 1 + Subscript[b, 2] - Subscript[b, 3]}, z] && !Element[Subscript[b, 1], Integers] && !Element[Subscript[b, 2], Integers] && !Element[Subscript[b, 3], Integers] && !Element[Subscript[b, 1] - Subscript[b, 2], Integers] && !Element[Subscript[b, 1] - Subscript[b, 3], Integers] && !Element[Subscript[b, 2] - Subscript[b, 3], 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,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]