html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 HypergeometricPFQ

 http://functions.wolfram.com/07.26.13.0002.01

 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] (MeijerG[{{1 - Subscript[a, 1], 1 - Subscript[a, 2]}, {}}, {{0, 1 - Subscript[b, 1]}, {1 - Subscript[b, 2], 1 - Subscript[b, 3]}}, z] + MeijerG[{{1 - Subscript[a, 1], 1 - Subscript[a, 2]}, {}}, {{0, 1 - Subscript[b, 2]}, {1 - Subscript[b, 1], 1 - Subscript[b, 3]}}, z] + MeijerG[{{1 - Subscript[a, 1], 1 - Subscript[a, 2]}, {}}, {{0, 1 - Subscript[b, 3]}, {1 - Subscript[b, 1], 1 - Subscript[b, 2]}}, z]) + Subscript[c, 3] (MeijerG[{{1 - Subscript[a, 1], 1 - Subscript[a, 2]}, {}}, {{0, 1 - Subscript[b, 1], 1 - Subscript[b, 2]}, {1 - Subscript[b, 3]}}, -z] + MeijerG[{{1 - Subscript[a, 1], 1 - Subscript[a, 2]}, {}}, {{0, 1 - Subscript[b, 1], 1 - Subscript[b, 3]}, {1 - Subscript[b, 2]}}, -z] + MeijerG[{{1 - Subscript[a, 1], 1 - Subscript[a, 2]}, {}}, {{0, 1 - Subscript[b, 2], 1 - Subscript[b, 3]}, {1 - Subscript[b, 1]}}, -z]) + Subscript[c, 4] MeijerG[{{1 - Subscript[a, 1], 1 - Subscript[a, 2]}, {}}, {{0, 1 - Subscript[b, 1], 1 - Subscript[b, 2], 1 - Subscript[b, 3]}, {}}, z]

 Standard Form

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 MathML Form

 z 3 w ( 4 ) TagBox[RowBox[List["(", "4", ")"]], Derivative] ( z ) + ( b 1 + b 2 + b 3 + 3 ) z 2 w ( 3 ) TagBox[RowBox[List["(", "3", ")"]], Derivative] ( z ) + ( b 1 + b 1 b 2 + b 2 + b 1 b 3 + b 2 b 3 + b 3 + 1 - z ) z w ′′ ( z ) + ( b 1 b 2 b 3 - ( a 1 + a 2 + 1 ) z ) w ( z ) - a 1 a 2 w ( z ) 0 /; w ( z ) c 1 2 F ~ 3 ( a 1 , a 2 ; b 1 , b 2 , b 3 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox[OverscriptBox["F", "~"], "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "2"], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["b", "1"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "3"], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox["z", HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] + c 2 ( G 2 , 4 2 , 2 ( z 1 - a 1 , 1 - a 2 0 , 1 - b 1 , 1 - b 2 , 1 - b 3 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "4"]], RowBox[List["2", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "2"]]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "2"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "3"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] + G 2 , 4 2 , 2 ( z 1 - a 1 , 1 - a 2 0 , 1 - b 2 , 1 - b 1 , 1 - b 3 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "4"]], RowBox[List["2", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "2"]]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "2"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "3"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] + G 2 , 4 2 , 2 ( z 1 - a 1 , 1 - a 2 0 , 1 - b 3 , 1 - b 1 , 1 - b 2 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "4"]], RowBox[List["2", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "2"]]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "3"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "2"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] ) + c 3 ( G 2 , 4 3 , 2 ( - z 1 - a 1 , 1 - a 2 0 , 1 - b 1 , 1 - b 2 , 1 - b 3 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "4"]], RowBox[List["3", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["-", "z"]], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "2"]]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "2"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "3"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] + G 2 , 4 3 , 2 ( - z 1 - a 1 , 1 - a 2 0 , 1 - b 1 , 1 - b 3 , 1 - b 2 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "4"]], RowBox[List["3", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["-", "z"]], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "2"]]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "3"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "2"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] + G 2 , 4 3 , 2 ( - z 1 - a 1 , 1 - a 2 0 , 1 - b 2 , 1 - b 3 , 1 - b 1 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "4"]], RowBox[List["3", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["-", "z"]], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "2"]]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "2"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "3"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "1"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] ) + c 4 G 2 , 4 4 , 2 ( z 1 - a 1 , 1 - a 2 0 , 1 - b 1 , 1 - b 2 , 1 - b 3 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "4"]], RowBox[List["4", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "2"]]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "2"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "3"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] Condition z 3 z 4 w z Subscript b 1 Subscript b 2 Subscript b 3 3 z 2 z 3 w z Subscript b 1 Subscript b 1 Subscript b 2 Subscript b 2 Subscript b 1 Subscript b 3 Subscript b 2 Subscript b 3 Subscript b 3 1 -1 z z z 2 w z Subscript b 1 Subscript b 2 Subscript b 3 -1 Subscript a 1 Subscript a 2 1 z z w z -1 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 MeijerG 1 -1 Subscript a 1 1 -1 Subscript a 2 0 1 -1 Subscript b 1 1 -1 Subscript b 2 1 -1 Subscript b 3 z MeijerG 1 -1 Subscript a 1 1 -1 Subscript a 2 0 1 -1 Subscript b 2 1 -1 Subscript b 1 1 -1 Subscript b 3 z MeijerG 1 -1 Subscript a 1 1 -1 Subscript a 2 0 1 -1 Subscript b 3 1 -1 Subscript b 1 1 -1 Subscript b 2 z Subscript c 3 MeijerG 1 -1 Subscript a 1 1 -1 Subscript a 2 0 1 -1 Subscript b 1 1 -1 Subscript b 2 1 -1 Subscript b 3 -1 z MeijerG 1 -1 Subscript a 1 1 -1 Subscript a 2 0 1 -1 Subscript b 1 1 -1 Subscript b 3 1 -1 Subscript b 2 -1 z MeijerG 1 -1 Subscript a 1 1 -1 Subscript a 2 0 1 -1 Subscript b 2 1 -1 Subscript b 3 1 -1 Subscript b 1 -1 z Subscript c 4 MeijerG 1 -1 Subscript a 1 1 -1 Subscript a 2 0 1 -1 Subscript b 1 1 -1 Subscript b 2 1 -1 Subscript b 3 z [/itex]

 Rule Form

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

 2007-05-02