Generalized hypergeometric function 4F3: Differential equations (formula 07.28.13.0007)

HypergeometricPFQ

$Mathematica Notation$

Hypergeometric Functions

HypergeometricPFQ[{a₁,a₂,a₃,a₄},{b₁,b₂,b₃},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/07.28.13.0007.01

Input Form

Derivative[4][w][z] + (((6 + Subscript[a, 1] + Subscript[a, 2] + Subscript[a, 3] + Subscript[a, 4]) Derivative[1][g][z])/(-1 + g[z]) - ((3 + Subscript[b, 1] + Subscript[b, 2] + Subscript[b, 3]) Derivative[1][g][z])/((-1 + g[z]) g[z]) - (6 Derivative[2][g][z])/ Derivative[1][g][z]) Derivative[3][w][z] + (((7 + Subscript[a, 1] Subscript[a, 2] + Subscript[a, 1] Subscript[a, 3] + Subscript[a, 2] Subscript[a, 3] + Subscript[a, 1] Subscript[a, 4] + Subscript[a, 2] Subscript[a, 4] + Subscript[a, 3] Subscript[a, 4] + 3 (Subscript[a, 1] + Subscript[a, 2] + Subscript[a, 3] + Subscript[a, 4])) Derivative[1][g][z]^2)/((-1 + g[z]) g[z]) - (((1 + Subscript[b, 2]) (1 + Subscript[b, 3]) + Subscript[b, 1] (1 + Subscript[b, 2] + Subscript[b, 3])) Derivative[1][g][z]^2)/ ((-1 + g[z]) g[z]^2) - (3 (6 + Subscript[a, 1] + Subscript[a, 2] + Subscript[a, 3] + Subscript[a, 4]) Derivative[2][g][z])/(-1 + g[z]) + (3 (3 + Subscript[b, 1] + Subscript[b, 2] + Subscript[b, 3]) Derivative[2][g][z])/((-1 + g[z]) g[z]) + (15 Derivative[2][g][z]^2)/ Derivative[1][g][z]^2 - (4 Derivative[3][g][z])/Derivative[1][g][z]) Derivative[2][w][z] + ((Subscript[b, 1] Subscript[b, 2] Subscript[b, 3] Derivative[1][g][z]^3)/ ((1 - g[z]) g[z]^3) + (3 (6 + Subscript[a, 1] + Subscript[a, 2] + Subscript[a, 3] + Subscript[a, 4]) Derivative[2][g][z]^2)/ ((-1 + g[z]) Derivative[1][g][z]) - (3 (3 + Subscript[b, 1] + Subscript[b, 2] + Subscript[b, 3]) Derivative[2][g][z]^2)/((-1 + g[z]) g[z] Derivative[1][g][z]) - (15 Derivative[2][g][z]^3)/Derivative[1][g][z]^3 + (1/((-1 + g[z]) g[z]^2)) (Derivative[1][g][z] ((1 + Subscript[a, 1] + Subscript[a, 2] + Subscript[a, 1] Subscript[a, 2] + Subscript[a, 3] + Subscript[a, 1] Subscript[a, 3] + Subscript[a, 2] Subscript[a, 3] + Subscript[a, 1] Subscript[a, 2] Subscript[a, 3] + Subscript[a, 4] + Subscript[a, 1] Subscript[a, 4] + Subscript[a, 2] Subscript[a, 4] + Subscript[a, 1] Subscript[a, 2] Subscript[a, 4] + Subscript[a, 3] Subscript[a, 4] + Subscript[a, 1] Subscript[a, 3] Subscript[a, 4] + Subscript[a, 2] Subscript[a, 3] Subscript[a, 4]) Derivative[1][g][z]^2 + ((1 + Subscript[b, 2]) (1 + Subscript[b, 3]) + Subscript[b, 1] (1 + Subscript[b, 2] + Subscript[b, 3])) Derivative[2][g][z])) - ((6 + Subscript[a, 1] + Subscript[a, 2] + Subscript[a, 3] + Subscript[a, 4]) Derivative[3][g][z])/(-1 + g[z]) + (10 Derivative[2][g][z] Derivative[3][g][z])/Derivative[1][g][z]^2 + (1/((-1 + g[z]) g[z])) ((-7 - Subscript[a, 1] Subscript[a, 2] - Subscript[a, 1] Subscript[a, 3] - Subscript[a, 2] Subscript[a, 3] - Subscript[a, 1] Subscript[a, 4] - Subscript[a, 2] Subscript[a, 4] - Subscript[a, 3] Subscript[a, 4] - 3 (Subscript[a, 1] + Subscript[a, 2] + Subscript[a, 3] + Subscript[a, 4])) Derivative[1][g][z] Derivative[2][g][z] + (3 + Subscript[b, 1] + Subscript[b, 2] + Subscript[b, 3]) Derivative[3][g][z]) - Derivative[4][g][z]/Derivative[1][g][z]) Derivative[1][w][z] + ((Subscript[a, 1] Subscript[a, 2] Subscript[a, 3] Subscript[a, 4] Derivative[1][g][z]^4)/((-1 + g[z]) g[z]^3)) w[z] == 0 /; w[z] == Subscript[c, 1] HypergeometricPFQRegularized[ {Subscript[a, 1], Subscript[a, 2], Subscript[a, 3], Subscript[a, 4]}, {Subscript[b, 1], Subscript[b, 2], Subscript[b, 3]}, g[z]] + Subscript[c, 2] (MeijerG[{{1 - Subscript[a, 1], 1 - Subscript[a, 2], 1 - Subscript[a, 3], 1 - Subscript[a, 4]}, {}}, {{0, 1 - Subscript[b, 1]}, {1 - Subscript[b, 2], 1 - Subscript[b, 3]}}, g[z]] + MeijerG[{{1 - Subscript[a, 1], 1 - Subscript[a, 2], 1 - Subscript[a, 3], 1 - Subscript[a, 4]}, {}}, {{0, 1 - Subscript[b, 2]}, {1 - Subscript[b, 1], 1 - Subscript[b, 3]}}, g[z]] + MeijerG[{{1 - Subscript[a, 1], 1 - Subscript[a, 2], 1 - Subscript[a, 3], 1 - Subscript[a, 4]}, {}}, {{0, 1 - Subscript[b, 3]}, {1 - Subscript[b, 1], 1 - Subscript[b, 2]}}, g[z]]) + Subscript[c, 3] (MeijerG[{{1 - Subscript[a, 1], 1 - Subscript[a, 2], 1 - Subscript[a, 3], 1 - Subscript[a, 4]}, {}}, {{0, 1 - Subscript[b, 1], 1 - Subscript[b, 2]}, {1 - Subscript[b, 3]}}, -g[z]] + MeijerG[{{1 - Subscript[a, 1], 1 - Subscript[a, 2], 1 - Subscript[a, 3], 1 - Subscript[a, 4]}, {}}, {{0, 1 - Subscript[b, 1], 1 - Subscript[b, 3]}, {1 - Subscript[b, 2]}}, -g[z]] + MeijerG[{{1 - Subscript[a, 1], 1 - Subscript[a, 2], 1 - Subscript[a, 3], 1 - Subscript[a, 4]}, {}}, {{0, 1 - Subscript[b, 2], 1 - Subscript[b, 3]}, {1 - Subscript[b, 1]}}, -g[z]]) + Subscript[c, 4] MeijerG[{{1 - Subscript[a, 1], 1 - Subscript[a, 2], 1 - Subscript[a, 3], 1 - Subscript[a, 4]}, {}}, {{0, 1 - Subscript[b, 1], 1 - Subscript[b, 2], 1 - Subscript[b, 3]}, {}}, g[z]]

Standard Form

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TagBox[RowBox[List["(", "3", ")"]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <msup> <mrow> <msup> <mi> g </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> g </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> g </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁢ </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> </mrow> <mo> - </mo> <mrow> <msub> <mi> a </mi> <mn> 3 </mn> </msub> <mo> ⁢ </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> </mrow> <mo> - </mo> <mrow> <msub> <mi> a </mi> <mn> 4 </mn> </msub> <mo> ⁢ </mo> 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</mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> g </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> - </mo> <mfrac> <mrow> <mn> 15 </mn> <mo> ⁢ </mo> <msup> <mrow> <msup> <mi> g </mi> <mi> ′′ </mi> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> <msup> <mrow> <msup> <mi> g </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mfrac> <mo> - </mo> <mfrac> <mrow> <msup> <mi> g </mi> <semantics> <mrow> <mo> ( </mo> <mn> 4 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", "4", ")"]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mrow> <msup> <mi> g </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> w </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> ⁢ </mo> <msub> <mi> a </mi> <mn> 3 </mn> </msub> <mo> ⁢ </mo> <msub> <mi> a </mi> <mn> 4 </mn> </msub> <mo> ⁢ </mo> <msup> <mrow> <msup> <mi> g </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> g </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mi> g </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mi> w </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo>  </mo> <mn> 0 </mn> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> w </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo>  </mo> <mrow> <mrow> <msub> <mi> c </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 4 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 3 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> a </mi> <mn> 3 </mn> </msub> <mo> , </mo> <msub> <mi> a </mi> <mn> 4 </mn> </msub> </mrow> <mo> ; </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> b </mi> <mn> 3 </mn> </msub> </mrow> <mo> ; </mo> <mrow> <mi> g </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "4"], SubscriptBox[OverscriptBox["F", "~"], "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], 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</bvar> <apply> <ci> g </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <partialdiff /> <bvar> <ci> z </ci> </bvar> <apply> <ci> g </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <partialdiff /> <bvar> <ci> z </ci> </bvar> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <apply> <partialdiff /> <bvar> <ci> z </ci> </bvar> <apply> <ci> g </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <ci> g </ci> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <ci> g </ci> <ci> z </ci> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <eq /> <apply> <ci> w </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </list> <list> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript 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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,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]