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; }

 HypergeometricPFQRegularized

 http://functions.wolfram.com/07.32.13.0003.01

 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] + Sum[Subscript[c, k + 1] z^(1 - Subscript[b, k]) HypergeometricPFQRegularized[{1 + Subscript[a, 1] - Subscript[b, k], \[Ellipsis], 1 + Subscript[a, p] - Subscript[b, k]}, {2 - Subscript[b, k], 1 + Subscript[b, 1] - Subscript[b, k], \[Ellipsis], 1 + Subscript[b, k - 1] - Subscript[b, k], 1 + Subscript[b, k + 1] - Subscript[b, k], \[Ellipsis], 1 + Subscript[b, q] - Subscript[b, k]}, z], {k, 1, q}] /; ForAll[{j, k}, Element[{j, k}, Integers] && j != k && 1 <= j <= q && 1 <= k <= q, !Element[Subscript[b, j] - Subscript[b, k], Integers]] && !Element[Subscript[b, k], Integers])

 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] + k = 1 q c k + 1 z 1 - b k p F ~ q ( a 1 - b k + 1 , , a p - b k + 1 ; 2 - b k , b 1 - b k + 1 , , b k - 1 - b k + 1 , b k + 1 - b k + 1 , , b q - b k + 1 ; 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[RowBox[List[SubscriptBox["a", "1"], "-", SubscriptBox["b", "k"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "p"], "-", SubscriptBox["b", "k"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["2", "-", SubscriptBox["b", "k"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["b", "1"], "-", SubscriptBox["b", "k"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["b", RowBox[List["k", "-", "1"]]], "-", SubscriptBox["b", "k"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["b", RowBox[List["k", "+", "1"]]], "-", SubscriptBox["b", "k"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["b", "q"], "-", SubscriptBox["b", "k"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox["z", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] /; { j , k } , { j , k } TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] j k 1 j q 1 k q ( b j - b k TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] ) b k TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] ) Condition z q z q 1 w z z q -1 q q -1 2 -1 k 1 q Subscript b k z q w z -1 z p z p w z -1 z p -1 p p -1 2 -1 l 1 p Subscript a l z p -1 w z d d z -1 k 1 q z d d z -1 Subscript b k -1 w z -1 l 1 p z d d z -1 Subscript a l w z -1 z q z q 1 w z -1 z q -1 q q -1 2 -1 k 1 q Subscript b k z q w z z p z p w z z p -1 p p -1 2 -1 k 1 p Subscript a k z p -1 w z w z l 1 p Subscript a l -1 w z l 1 p Subscript a l 0 Condition w z Subscript c 1 HypergeometricPFQ Subscript a 1 Subscript a p Subscript b 1 Subscript b q z k 1 q Subscript c k 1 z 1 -1 Subscript b k HypergeometricPFQ Subscript a 1 -1 Subscript b k 1 Subscript a p -1 Subscript b k 1 2 -1 Subscript b k Subscript b 1 -1 Subscript b k 1 Subscript b k -1 -1 Subscript b k 1 Subscript b k 1 -1 Subscript b k 1 Subscript b q -1 Subscript b k 1 z j k j k j k 1 j q 1 k q Subscript b j -1 Subscript b k Subscript b k [/itex]

 Rule Form

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

 2007-05-02