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 HypergeometricPFQ

 http://functions.wolfram.com/07.27.13.0009.01

 Input Form

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

 Standard Form

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 MathML Form

 w ( 3 ) TagBox[RowBox[List["(", "3", ")"]], Derivative] ( z ) + ( ( a 1 + a 2 + a 3 + 3 ) g ( z ) g ( z ) - 1 - ( b 1 + b 2 + 1 ) g ( z ) ( g ( z ) - 1 ) g ( z ) - 3 h ( z ) h ( z ) - 3 g ′′ ( z ) g ( z ) ) w ′′ ( z ) + ( b 1 b 2 g ( z ) 2 ( 1 - g ( z ) ) g ( z ) 2 + 2 ( b 1 + b 2 + 1 ) h ( z ) g ( z ) ( g ( z ) - 1 ) g ( z ) h ( z ) - 2 ( a 1 + a 2 + a 3 + 3 ) h ( z ) g ( z ) ( g ( z ) - 1 ) h ( z ) + 6 h ( z ) 2 h ( z ) 2 + 3 g ′′ ( z ) 2 g ( z ) 2 + 6 h ( z ) g ′′ ( z ) h ( z ) g ( z ) + ( ( a 2 + 1 ) ( a 3 + 1 ) + a 1 ( a 2 + a 3 + 1 ) ) g ( z ) 2 + ( b 1 + b 2 + 1 ) g ′′ ( z ) ( g ( z ) - 1 ) g ( z ) - ( a 1 + a 2 + a 3 + 3 ) g ′′ ( z ) g ( z ) - 1 - 3 h ′′ ( z ) h ( z ) - g ( 3 ) TagBox[RowBox[List["(", "3", ")"]], Derivative] ( z ) g ( z ) ) w ( z ) + w ( z ) ( - a 1 a 2 a 3 g ( z ) 3 ( 1 - g ( z ) ) g ( z ) 2 - b 1 b 2 h ( z ) g ( z ) 2 ( 1 - g ( z ) ) g ( z ) 2 h ( z ) + 2 ( a 1 + a 2 + a 3 + 3 ) h ( z ) 2 g ( z ) ( g ( z ) - 1 ) h ( z ) 2 - 2 ( b 1 + b 2 + 1 ) h ( z ) 2 g ( z ) ( g ( z ) - 1 ) g ( z ) h ( z ) 2 + 6 h ( z ) h ′′ ( z ) h ( z ) 2 + 1 ( g ( z ) - 1 ) g ( z ) h ( z ) ( ( b 1 + b 2 + 1 ) g ( z ) h ′′ ( z ) - h ( z ) ( ( ( a 2 + 1 ) ( a 3 + 1 ) + a 1 ( a 2 + a 3 + 1 ) ) g ( z ) 2 + ( b 1 + b 2 + 1 ) g ′′ ( z ) ) ) + ( a 1 + a 2 + a 3 + 3 ) ( h ( z ) g ′′ ( z ) - g ( z ) h ′′ ( z ) ) ( g ( z ) - 1 ) h ( z ) + 3 g ′′ ( z ) h ′′ ( z ) + h ( z ) g ( 3 ) TagBox[RowBox[List["(", "3", ")"]], Derivative] ( z ) h ( z ) g ( z ) - h ( 3 ) TagBox[RowBox[List["(", "3", ")"]], Derivative] ( z ) h ( z ) - 6 h ( z ) 3 h ( z ) 3 - 6 h ( z ) 2 g ′′ ( z ) h ( z ) 2 g ( z ) - 3 h ( z ) g ′′ ( z ) 2 h ( z ) g ( z ) 2 ) 0 /; w ( z ) c 1 h ( z ) 3 F ~ 2 ( a 1 , a 2 , a 3 ; b 1 , b 2 ; g ( z ) ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "3"], SubscriptBox[OverscriptBox["F", "~"], "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[SubscriptBox["a", "2"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[SubscriptBox["a", "3"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["b", "1"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[SubscriptBox["b", "2"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[RowBox[List["g", "(", "z", ")"]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQRegularized] + c 2 h ( z ) ( G 3 , 3 2 , 3 ( g ( z ) 1 - a 1 , 1 - a 2 , 1 - a 3 0 , 1 - b 1 , 1 - b 2 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["3", ",", "3"]], RowBox[List["2", ",", "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["g", "(", "z", ")"]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "1"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "2"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "3"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "1"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "2"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False], Rule[Selectable, False]] + G 3 , 3 2 , 3 ( g ( z ) 1 - 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g ( z ) 1 - a 1 , 1 - a 2 , 1 - a 3 0 , 1 - b 1 , 1 - b 2 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["3", ",", "3"]], RowBox[List["3", ",", "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["-", RowBox[List["g", "(", "z", ")"]]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "1"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "2"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "3"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "1"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "2"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False], Rule[Selectable, False]] Condition z 3 w z Subscript a 1 Subscript a 2 Subscript a 3 3 z g z g z -1 -1 -1 Subscript b 1 Subscript b 2 1 z g z g z -1 g z -1 -1 3 z h z h z -1 -1 3 z 2 g z z g z -1 z 2 w z Subscript b 1 Subscript b 2 z g z 2 1 -1 g z g z 2 -1 2 Subscript b 1 Subscript b 2 1 z h z z g z g z -1 g z h z -1 -1 2 Subscript a 1 Subscript a 2 Subscript a 3 3 z h z z g z g z -1 h z -1 6 z h z 2 h z 2 -1 3 z 2 g z 2 z g z 2 -1 6 z h z z 2 g z h z z g z -1 Subscript a 2 1 Subscript a 3 1 Subscript a 1 Subscript a 2 Subscript a 3 1 z g z 2 Subscript b 1 Subscript b 2 1 z 2 g z g z -1 g z -1 -1 Subscript a 1 Subscript a 2 Subscript a 3 3 z 2 g z g z -1 -1 -1 3 z 2 h z h z -1 -1 z 3 g z z g z -1 z w z w z -1 Subscript a 1 Subscript a 2 Subscript a 3 z g z 3 1 -1 g z g z 2 -1 -1 Subscript b 1 Subscript b 2 z h z z g z 2 1 -1 g z g z 2 h z -1 2 Subscript a 1 Subscript a 2 Subscript a 3 3 z h z 2 z g z g z -1 h z 2 -1 -1 2 Subscript b 1 Subscript b 2 1 z h z 2 z g z g z -1 g z h z 2 -1 6 z h z z 2 h z h z 2 -1 1 g z -1 g z h z -1 Subscript b 1 Subscript b 2 1 z g z z 2 h z -1 z h z Subscript a 2 1 Subscript a 3 1 Subscript a 1 Subscript a 2 Subscript a 3 1 z g z 2 Subscript b 1 Subscript b 2 1 z 2 g z Subscript a 1 Subscript a 2 Subscript a 3 3 z h z z 2 g z -1 z g z z 2 h z g z -1 h z -1 3 z 2 g z z 2 h z z h z z 3 g z h z z g z -1 -1 z 3 h z h z -1 -1 6 z h z 3 h z 3 -1 -1 6 z h z 2 z 2 g z h z 2 z g z -1 -1 3 z h z z 2 g z 2 h z z g z 2 -1 0 w z Subscript c 1 h z HypergeometricPFQRegularized Subscript a 1 Subscript a 2 Subscript a 3 Subscript b 1 Subscript b 2 g z Subscript c 2 h z MeijerG 1 -1 Subscript a 1 1 -1 Subscript a 2 1 -1 Subscript a 3 0 1 -1 Subscript b 1 1 -1 Subscript b 2 g z MeijerG 1 -1 Subscript a 1 1 -1 Subscript a 2 1 -1 Subscript a 3 0 1 -1 Subscript b 2 1 -1 Subscript b 1 g z Subscript c 3 h z MeijerG 1 -1 Subscript a 1 1 -1 Subscript a 2 1 -1 Subscript a 3 0 1 -1 Subscript b 1 1 -1 Subscript b 2 -1 g z [/itex]

 Rule Form

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 Date Added to functions.wolfram.com (modification date)

 2007-05-02