Generalized hypergeometric function 3F2: Differential equations (formula 07.27.13.0011)

HypergeometricPFQ

$Mathematica Notation$

Hypergeometric Functions

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

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

Representation of fundamental system solutions near zero

http://functions.wolfram.com/07.27.13.0011.01

Input Form

Derivative[3][w][z] + (-((3 (-1 + r + s))/z) + (a r z^(-1 + r) (3 + Subscript[a, 1] + Subscript[a, 2] + Subscript[a, 3]))/(-1 + a z^r) - (r (1 + Subscript[b, 1] + Subscript[b, 2]))/(z (-1 + a z^r))) Derivative[2][w][z] + ((1 - 3 r + 2 r^2 - 3 s + 6 r s + 3 s^2)/z^2 + (r (-1 + r + 2 s))/(z^2 (-1 + a z^r)) + (a r z^(-2 + r) (3 - 2 r - 6 s + Subscript[a, 3] - 2 s Subscript[a, 3] + Subscript[a, 2] (1 - 2 s + r Subscript[a, 3]) + Subscript[a, 1] (1 - 2 s + r Subscript[a, 2] + r Subscript[a, 3])))/ (-1 + a z^r) - (r ((-(-1 + r + 2 s)) Subscript[b, 2] + Subscript[b, 1] (1 - r - 2 s + r Subscript[b, 2])))/ (z^2 (-1 + a z^r))) Derivative[1][w][z] + (-((s (2 r^2 + 3 r s + s^2))/z^3) - (r s (r + s))/(z^3 (-1 + a z^r)) + (a r z^(-3 + r) (Subscript[a, 1] (s - r Subscript[a, 2]) (s - r Subscript[a, 3]) + s (2 r + 3 s + s Subscript[a, 3] + Subscript[a, 2] (s - r Subscript[a, 3]))))/(-1 + a z^r) + (r s ((-(r + s)) Subscript[b, 2] + Subscript[b, 1] (-r - s + r Subscript[b, 2])))/(z^3 (-1 + a z^r))) w[z] == 0 /; w[z] == Subscript[c, 1] z^s HypergeometricPFQRegularized[ {Subscript[a, 1], Subscript[a, 2], Subscript[a, 3]}, {Subscript[b, 1], Subscript[b, 2]}, a z^r] + Subscript[c, 2] z^s (MeijerG[{{1 - Subscript[a, 1], 1 - Subscript[a, 2], 1 - Subscript[a, 3]}, {}}, {{0, 1 - Subscript[b, 1]}, {1 - Subscript[b, 2]}}, a z^r] + MeijerG[{{1 - Subscript[a, 1], 1 - Subscript[a, 2], 1 - Subscript[a, 3]}, {}}, {{0, 1 - Subscript[b, 2]}, {1 - Subscript[b, 1]}}, a z^r]) + Subscript[c, 3] z^s MeijerG[{{1 - Subscript[a, 1], 1 - Subscript[a, 2], 1 - Subscript[a, 3]}, {}}, {{0, 1 - Subscript[b, 1], 1 - Subscript[b, 2]}, {}}, (-a) z^r]

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,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]