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/07.31.13.0009.01

 Input Form

 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}] == 0 /; w[z] == Sum[(Subscript[c, k] HypergeometricPFQRegularized[ {Subscript[a, k], 1 + Subscript[a, k] - Subscript[b, 1], \[Ellipsis], 1 + Subscript[a, k] - Subscript[b, q]}, {1 + Subscript[a, k] - Subscript[a, 1], \[Ellipsis], 1 + Subscript[a, k] - Subscript[a, k - 1], 1 + Subscript[a, k] - Subscript[a, k + 1], \[Ellipsis], 1 + Subscript[a, k] - Subscript[a, p]}, (-1)^(1 - p + q)/z])/z^Subscript[a, k], {k, 1, p}] && ForAll[{j, k}, Element[{j, k}, Integers] && j != k && 1 <= j <= p && 1 <= k <= p, !Element[Subscript[a, j] - Subscript[a, k], Integers]]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", "z"], RowBox[List["Fold", "[", RowBox[List[RowBox[List["Function", "[", RowBox[List[RowBox[List["{", RowBox[List["f", ",", "k"]], "}"]], ",", RowBox[List[RowBox[List["z", " ", RowBox[List[SubscriptBox["\[PartialD]", "z"], "f"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["b", "k"], "-", "1"]], ")"]], " ", "f"]]]]]], "]"]], ",", RowBox[List["w", "[", "z", "]"]], ",", RowBox[List["{", RowBox[List["1", ",", "\[Ellipsis]", ",", "q"]], "}"]]]], "]"]]]], "-", " ", RowBox[List["Fold", "[", RowBox[List[RowBox[List["Function", "[", RowBox[List[RowBox[List["{", RowBox[List["f", ",", "l"]], "}"]], ",", RowBox[List[RowBox[List["z", " ", RowBox[List[SubscriptBox["\[PartialD]", "z"], "f"]]]], "+", RowBox[List[SubscriptBox["a", "l"], " ", "f"]]]]]], "]"]], ",", RowBox[List["w", "[", "z", "]"]], ",", RowBox[List["{", RowBox[List["1", ",", "\[Ellipsis]", ",", "p"]], "}"]]]], "]"]]]], "\[Equal]", "0"]], "/;", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "p"], RowBox[List[SubscriptBox["c", "k"], " ", SuperscriptBox["z", RowBox[List["-", SubscriptBox["a", "k"]]]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "k"], ",", RowBox[List["1", "+", SubscriptBox["a", "k"], "-", SubscriptBox["b", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["a", "k"], "-", SubscriptBox["b", "q"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "k"], "-", SubscriptBox["a", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["a", "k"], "-", SubscriptBox["a", RowBox[List["k", "-", "1"]]]]], ",", RowBox[List["1", "+", SubscriptBox["a", "k"], "-", SubscriptBox["a", RowBox[List["k", "+", "1"]]]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["a", "k"], "-", SubscriptBox["a", "p"]]]]], "}"]], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "-", "p", "+", "q"]]], "z"]]], "]"]]]]]]]], "\[And]", RowBox[List[SubscriptBox["\[ForAll]", RowBox[List[RowBox[List["{", RowBox[List["j", ",", "k"]], "}"]], ",", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List["j", ",", "k"]], "}"]], "\[Element]", "Integers"]], "\[And]", RowBox[List["j", "\[NotEqual]", "k"]], "\[And]", RowBox[List["1", "\[LessEqual]", "j", "\[LessEqual]", "p"]], "\[And]", RowBox[List["1", "\[LessEqual]", "k", "\[LessEqual]", "p"]]]]]]], RowBox[List["(", "\[InvisibleSpace]", RowBox[List["Not", "[", RowBox[List[RowBox[List[SubscriptBox["a", "j"], "-", SubscriptBox["a", "k"]]], "\[Element]", "Integers"]], "]"]], ")"]]]]]], ")"]]]]]]

 MathML Form

 ( d d z k = 1 q ( z d d z + b k - 1 ) - l = 1 p ( z d d z + a l ) ) w ( z ) 0 /; w ( z ) k = 1 p c k z - a k q + 1 F ~ p - 1 ( a k , a k - b 1 + 1 , , a k - b q + 1 ; 1 - a 1 + a k , , 1 - a k - 1 + a k , 1 - a k + 1 + a k , , 1 - a p + a k ; ( - 1 ) 1 - p + q z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["q", "+", "1"]], TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox[RowBox[List["p", "-", "1"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "k"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "k"], "-", SubscriptBox["b", "1"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "k"], "-", SubscriptBox["b", "q"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "1"], "+", SubscriptBox["a", "k"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", RowBox[List["k", "-", "1"]]], "+", SubscriptBox["a", "k"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", RowBox[List["k", "+", "1"]]], "+", SubscriptBox["a", "k"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "p"], "+", SubscriptBox["a", "k"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "-", "p", "+", "q"]]], "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 p 1 k p ( a j - a k TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] ) Condition d d z -1 k 1 q z d d z -1 Subscript b k -1 -1 l 1 p z d d z -1 Subscript a l w z 0 w z k 1 p Subscript c k z -1 Subscript a k HypergeometricPFQ Subscript a k Subscript a k -1 Subscript b 1 1 Subscript a k -1 Subscript b q 1 1 -1 Subscript a 1 Subscript a k 1 -1 Subscript a k -1 Subscript a k 1 -1 Subscript a k 1 Subscript a k 1 -1 Subscript a p Subscript a k -1 1 -1 p q z -1 j k j k j k 1 j p 1 k p Subscript a j -1 Subscript a k [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["z_"]]], RowBox[List["Fold", "[", RowBox[List[RowBox[List["Function", "[", RowBox[List[RowBox[List["{", RowBox[List["f_", ",", "k"]], "}"]], ",", RowBox[List[RowBox[List["z_", " ", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["z_"]]], "f_"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["b_", "k"], "-", "1"]], ")"]], " ", "f_"]]]]]], "]"]], ",", RowBox[List["w", "[", "z_", "]"]], ",", RowBox[List["{", RowBox[List["1", ",", "\[Ellipsis]_", ",", "q_"]], "}"]]]], "]"]]]], "-", RowBox[List["Fold", "[", RowBox[List[RowBox[List["Function", "[", RowBox[List[RowBox[List["{", RowBox[List["f_", ",", "l_"]], "}"]], ",", RowBox[List[RowBox[List["z_", " ", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["z_"]]], "f_"]]]], "+", RowBox[List[SubscriptBox["a_", "l_"], " ", "f_"]]]]]], "]"]], ",", RowBox[List["w", "[", "z_", "]"]], ",", RowBox[List["{", RowBox[List["1", ",", "\[Ellipsis]_", ",", "p_"]], "}"]]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List["0", "/;", RowBox[List[RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "p"], RowBox[List[SubscriptBox["c", "k"], " ", SuperscriptBox["z", RowBox[List["-", SubscriptBox["a", "k"]]]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "k"], ",", RowBox[List["1", "+", SubscriptBox["a", "k"], "-", SubscriptBox["b", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["a", "k"], "-", SubscriptBox["b", "q"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "k"], "-", SubscriptBox["a", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["a", "k"], "-", SubscriptBox["a", RowBox[List["k", "-", "1"]]]]], ",", RowBox[List["1", "+", SubscriptBox["a", "k"], "-", SubscriptBox["a", RowBox[List["k", "+", "1"]]]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["a", "k"], "-", SubscriptBox["a", "p"]]]]], "}"]], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "-", "p", "+", "q"]]], "z"]]], "]"]]]]]]]], "&&", RowBox[List[SubscriptBox["\[ForAll]", RowBox[List[RowBox[List["{", RowBox[List["j", ",", "k"]], "}"]], ",", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List["j", ",", "k"]], "}"]], "\[Element]", "Integers"]], "&&", RowBox[List["j", "\[NotEqual]", "k"]], "&&", RowBox[List["1", "\[LessEqual]", "j", "\[LessEqual]", "p"]], "&&", RowBox[List["1", "\[LessEqual]", "k", "\[LessEqual]", "p"]]]]]]], RowBox[List["(", RowBox[List["!", RowBox[List[RowBox[List[SubscriptBox["a", "j"], "-", SubscriptBox["a", "k"]]], "\[Element]", "Integers"]]]], ")"]]]]]]]]]]]]

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

 2007-05-02