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/07.31.13.0004.01

 Input Form

 z^q Derivative[q + 1][w][z] + z^(q - 1) ((q (q - 1))/2 + Sum[Subscript[b, k], {k, 1, q}]) Derivative[q][w][z] - z^p Derivative[p][w][z] - z^(p - 1) ((p (p - 1))/2 + Sum[Subscript[a, l], {l, 1, p}]) Derivative[p - 1][w][z] + (D[Fold[Function[{f, k}, z D[f, z] + (Subscript[b, k] - 1) f], w[z], {1, \[Ellipsis], q}], z] - Fold[Function[{f, l}, z D[f, z] + Subscript[a, l] f], w[z], {1, \[Ellipsis], p}] - z^q Derivative[q + 1][w][z] - z^(q - 1) ((q (q - 1))/2 + Sum[Subscript[b, k], {k, 1, q}]) Derivative[q][w][z] + z^p Derivative[p][w][z] + z^(p - 1) ((p (p - 1))/2 + Sum[Subscript[a, k], {k, 1, p}]) Derivative[p - 1][w][z] + w[z] Product[Subscript[a, l], {l, 1, p}]) - w[z] Product[Subscript[a, l], {l, 1, p}] == 0 /; w[z] == Subscript[c, 1] HypergeometricPFQRegularized[ {Subscript[a, 1], \[Ellipsis], Subscript[a, p]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, z] + Subscript[c, 2] Sum[MeijerG[{{1 - Subscript[a, 1], \[Ellipsis], 1 - Subscript[a, p]}, {}}, {{0, 1 - Subscript[b, k]}, {1 - Subscript[b, 1], \[Ellipsis], 1 - Subscript[b, k - 1], 1 - Subscript[b, k + 1], \[Ellipsis], 1 - Subscript[b, q]}}, z], {k, 1, q}] + \[Ellipsis] + Subscript[c, q] Sum[MeijerG[{{1 - Subscript[a, 1], \[Ellipsis], 1 - Subscript[a, p]}, {}}, {{0, 1 - Subscript[b, 1], \[Ellipsis], 1 - Subscript[b, k - 1], 1 - Subscript[b, k + 1], \[Ellipsis], 1 - Subscript[b, q]}, {1 - Subscript[b, k]}}, (-1)^q z], {k, 1, q}] + Subscript[c, q + 1] MeijerG[{{1 - Subscript[a, 1], \[Ellipsis], 1 - Subscript[a, p]}, {}}, {{0, 1 - Subscript[b, 1], \[Ellipsis], 1 - Subscript[b, q]}, {}}, (-1)^(q + 1) z]

 Standard Form

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 MathML Form

 z q w ( q + 1 ) TagBox[RowBox[List["(", RowBox[List["q", "+", "1"]], ")"]], Derivative] ( z ) + z q - 1 ( q ( q - 1 ) 2 + k = 1 q b k ) w ( q ) TagBox[RowBox[List["(", "q", ")"]], Derivative] ( z ) - z p w ( p ) TagBox[RowBox[List["(", "p", ")"]], Derivative] ( z ) - z p - 1 ( p ( p - 1 ) 2 + l = 1 p a l ) w ( p - 1 ) TagBox[RowBox[List["(", RowBox[List["p", "-", "1"]], ")"]], Derivative] ( z ) + ( ( d d z k = 1 q ( z d d z + b k - 1 ) ) w ( z ) - l = 1 p ( z d d z + a l ) w ( z ) - z q w ( q + 1 ) TagBox[RowBox[List["(", RowBox[List["q", "+", "1"]], ")"]], Derivative] ( z ) - z q - 1 ( q ( q - 1 ) 2 + k = 1 q b k ) w ( q ) TagBox[RowBox[List["(", "q", ")"]], Derivative] ( z ) + z p w ( p ) TagBox[RowBox[List["(", "p", ")"]], Derivative] ( z ) + z p - 1 ( p ( p - 1 ) 2 + k = 1 p a k ) w ( p - 1 ) TagBox[RowBox[List["(", RowBox[List["p", "-", "1"]], ")"]], Derivative] ( z ) + w ( z ) l = 1 p a l ) - w ( z ) l = 1 p a l 0 /; ( w ( z ) c 1 p F ~ q ( a 1 , , a p ; b 1 , , b q ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["p", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["q", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "p"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["b", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "q"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox["z", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] + c 2 k = 1 q G p , q + 1 2 , p ( z 1 - a 1 , , 1 - a p 0 , 1 - b k , 1 - b 1 , , 1 - b k - 1 , 1 - b k + 1 , , 1 - b q ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["p", ",", RowBox[List["q", "+", "1"]]]], RowBox[List["2", ",", "p"]]], "\[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["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "p"]]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "k"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", RowBox[List["k", "-", "1"]]]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", RowBox[List["k", "+", "1"]]]]], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "q"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, True]] + + c q k = 1 q G p , q + 1 q , p ( ( - 1 ) q z 1 - a 1 , , 1 - a p 0 , 1 - b 1 , , 1 - b k - 1 , 1 - b k + 1 , , 1 - b q , 1 - b k ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["p", ",", RowBox[List["q", "+", "1"]]]], RowBox[List["q", ",", "p"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "q"], " ", "z"]], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "p"]]], 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["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", RowBox[List["k", "-", "1"]]]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", RowBox[List["k", "+", "1"]]]]], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "q"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "k"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, True]] + c q + 1 G p , q + 1 q + 1 , p ( ( - 1 ) q + 1 z 1 - a 1 , , 1 - a p 0 , 1 - b 1 , , 1 - b q ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["p", ",", RowBox[List["q", "+", "1"]]]], RowBox[List[RowBox[List["q", "+", "1"]], ",", "p"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["q", "+", "1"]]], " ", "z"]], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "p"]]], 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["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "q"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, True]] ) FormBox RowBox RowBox RowBox RowBox SuperscriptBox z q RowBox SuperscriptBox w TagBox RowBox ( RowBox q + 1 ) Derivative ( z ) + RowBox SuperscriptBox z RowBox q - 1 RowBox ( RowBox FractionBox RowBox q RowBox ( RowBox q - 1 ) 2 + RowBox UnderoverscriptBox RowBox k = 1 q SubscriptBox b k ) RowBox SuperscriptBox w TagBox RowBox ( q ) Derivative ( z ) - RowBox SuperscriptBox z p RowBox SuperscriptBox w TagBox RowBox ( p ) Derivative ( z ) - RowBox SuperscriptBox z RowBox p - 1 RowBox ( RowBox FractionBox RowBox p RowBox ( RowBox p - 1 ) 2 + RowBox UnderoverscriptBox RowBox l = 1 p SubscriptBox a l ) RowBox SuperscriptBox w TagBox RowBox ( RowBox p - 1 ) Derivative ( z ) + RowBox ( RowBox RowBox RowBox ( RowBox FractionBox d RowBox d z RowBox UnderoverscriptBox RowBox k = 1 q RowBox ( RowBox RowBox z FractionBox d RowBox d z + SubscriptBox b k - 1 ) ) RowBox w ( z ) - RowBox UnderoverscriptBox RowBox l = 1 p RowBox RowBox ( RowBox RowBox z FractionBox d RowBox d z + SubscriptBox a l ) RowBox w ( z ) - RowBox SuperscriptBox z q RowBox SuperscriptBox w TagBox RowBox ( RowBox q + 1 ) Derivative ( z ) - RowBox SuperscriptBox z RowBox q - 1 RowBox ( RowBox FractionBox RowBox q RowBox ( RowBox q - 1 ) 2 + RowBox UnderoverscriptBox RowBox k = 1 q SubscriptBox b k ) RowBox SuperscriptBox w TagBox RowBox ( q ) Derivative ( z ) + RowBox SuperscriptBox z p RowBox SuperscriptBox w TagBox RowBox ( p ) Derivative ( z ) + RowBox SuperscriptBox z RowBox p - 1 RowBox ( RowBox FractionBox RowBox p RowBox ( RowBox p - 1 ) 2 + RowBox UnderoverscriptBox RowBox k = 1 p SubscriptBox a k ) RowBox SuperscriptBox w TagBox RowBox ( RowBox p - 1 ) Derivative ( z ) + RowBox RowBox w ( z ) RowBox UnderoverscriptBox RowBox l = 1 p SubscriptBox a l ) - RowBox RowBox w ( z ) RowBox UnderoverscriptBox RowBox l = 1 p SubscriptBox a l 0 /; RowBox ( RowBox RowBox w ( z ) RowBox RowBox SubscriptBox c 1 TagBox TagBox RowBox RowBox SubscriptBox FormBox p TraditionalForm SubscriptBox OverscriptBox F ~ FormBox q TraditionalForm RowBox ( RowBox TagBox TagBox RowBox TagBox SubscriptBox a 1 HypergeometricPFQ Rule Editable , TagBox HypergeometricPFQ Rule Editable , TagBox SubscriptBox a p HypergeometricPFQ Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable ; TagBox TagBox RowBox TagBox SubscriptBox b 1 HypergeometricPFQ Rule Editable , TagBox HypergeometricPFQ Rule Editable , TagBox SubscriptBox b q HypergeometricPFQ Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable ; TagBox z HypergeometricPFQ Rule Editable ) InterpretTemplate Function HypergeometricPFQ Slot 1 Slot 2 Slot 3 Rule Editable HypergeometricPFQ + RowBox SubscriptBox c 2 RowBox UnderoverscriptBox RowBox k = 1 q ErrorBox TagBox RowBox SubsuperscriptBox TagBox G MeijerG RowBox p , RowBox q + 1 RowBox 2 , p RowBox ( RowBox TagBox z MeijerG Rule Editable GridBox RowBox TagBox RowBox 1 - SubscriptBox a 1 MeijerG Rule Editable , TagBox MeijerG Rule Editable , TagBox RowBox 1 - SubscriptBox a p MeijerG Rule Editable RowBox TagBox 0 MeijerG Rule Editable , TagBox RowBox 1 - SubscriptBox b k MeijerG Rule Editable , TagBox RowBox 1 - SubscriptBox b 1 MeijerG Rule Editable , TagBox MeijerG Rule Editable , TagBox RowBox 1 - SubscriptBox b RowBox k - 1 MeijerG Rule Editable , TagBox RowBox 1 - SubscriptBox b RowBox k + 1 MeijerG Rule Editable , TagBox MeijerG Rule Editable , TagBox RowBox 1 - SubscriptBox b q MeijerG Rule Editable ) MeijerG Rule Editable + + RowBox SubscriptBox c q RowBox UnderoverscriptBox RowBox k = 1 q TagBox RowBox SubsuperscriptBox TagBox G MeijerG RowBox p , RowBox q + 1 RowBox q , p RowBox ( RowBox TagBox RowBox SuperscriptBox RowBox ( RowBox - 1 ) q z MeijerG Rule Editable GridBox RowBox TagBox RowBox 1 - SubscriptBox a 1 MeijerG Rule Editable , TagBox MeijerG Rule Editable , TagBox RowBox 1 - SubscriptBox a p MeijerG Rule Editable RowBox TagBox 0 MeijerG Rule Editable , TagBox RowBox 1 - SubscriptBox b 1 MeijerG Rule Editable , TagBox MeijerG Rule Editable , TagBox RowBox 1 - SubscriptBox b RowBox k - 1 MeijerG Rule Editable , TagBox RowBox 1 - SubscriptBox b RowBox k + 1 MeijerG Rule Editable , TagBox MeijerG Rule Editable , TagBox RowBox 1 - SubscriptBox b q MeijerG Rule Editable , TagBox RowBox 1 - SubscriptBox b k MeijerG Rule Editable ) MeijerG Rule Editable + RowBox SubscriptBox c RowBox q + 1 TagBox RowBox SubsuperscriptBox TagBox G MeijerG RowBox p , RowBox q + 1 RowBox RowBox q + 1 , p RowBox ( RowBox TagBox RowBox SuperscriptBox RowBox ( RowBox - 1 ) RowBox q + 1 z MeijerG Rule Editable GridBox RowBox TagBox RowBox 1 - SubscriptBox a 1 MeijerG Rule Editable , TagBox MeijerG Rule Editable , TagBox RowBox 1 - SubscriptBox a p MeijerG Rule Editable RowBox TagBox 0 MeijerG Rule Editable , TagBox RowBox 1 - SubscriptBox b 1 MeijerG Rule Editable , TagBox MeijerG Rule Editable , TagBox RowBox 1 - SubscriptBox b q MeijerG Rule Editable ) MeijerG Rule Editable ) TraditionalForm [/itex]

 Rule Form

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

 2007-05-02