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/07.32.06.0049.01

 Input Form

 HypergeometricPFQRegularized[{Subscript[a, 1], \[Ellipsis], Subscript[a, q + 1]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, z] == (Pi^q/Product[Gamma[Subscript[a, k]], {k, 1, q + 1}]) Sum[(((Gamma[Subscript[a, k] + u] Product[If[j == k, 1, Csc[Pi (Subscript[a, j] - Subscript[a, k])]], {j, 1, q + 1}])/ ((-x)^u Product[Gamma[Subscript[b, j] - Subscript[a, k]], {j, 1, q}])) HypergeometricPFQRegularized[{Subscript[a, k] + u, 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, q + 1]}, 1/x] ((z - x)^u/u!))/ (E^(Pi I Subscript[a, k] (1 + 2 Floor[Arg[x - z]/(2 Pi)])) x^Subscript[a, k]), {k, 1, q + 1}, {u, 0, Infinity}] /; ForAll[{j, k}, Element[{j, k}, Integers] && j != k && 1 <= j <= q + 1 && 1 <= k <= q + 1, !Element[Subscript[a, j] - Subscript[a, k], Integers]] && x > 1

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["q", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[" ", SuperscriptBox["\[Pi]", "q"]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], RowBox[List["q", "+", "1"]]], RowBox[List["Gamma", "[", SubscriptBox["a", "k"], "]"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["q", "+", "1"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["u", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "x"]], ")"]], RowBox[List["-", "u"]]], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["a", "k"], "+", "u"]], "]"]], " ", RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], RowBox[List["q", "+", "1"]]], RowBox[List["If", "[", RowBox[List[RowBox[List["j", "\[Equal]", "k"]], ",", "1", ",", RowBox[List["Csc", "[", RowBox[List["\[Pi]", RowBox[List["(", RowBox[List[SubscriptBox["a", "j"], "-", SubscriptBox["a", "k"]]], ")"]]]], "]"]]]], "]"]]]], ")"]], " "]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "q"], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "j"], "-", SubscriptBox["a", "k"]]], "]"]]]]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", " ", "\[Pi]"]], " ", "\[ImaginaryI]", " ", SubscriptBox["a", "k"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["x", "-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]], ")"]]]]], SuperscriptBox["x", RowBox[List["-", SubscriptBox["a", "k"]]]], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["a", "k"], "+", "u"]], ",", 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", RowBox[List["q", "+", "1"]]]]]]], "}"]], ",", FractionBox["1", "x"]]], "]"]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "u"], RowBox[List["u", "!"]]]]]]]]]]]]], "/;", RowBox[List[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]", RowBox[List["q", "+", "1"]]]], "\[And]", RowBox[List["1", "\[LessEqual]", "k", "\[LessEqual]", RowBox[List["q", "+", "1"]]]]]]]]], RowBox[List["(", "\[InvisibleSpace]", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["a", "j"], "-", SubscriptBox["a", "k"]]], "\[Element]", "Integers"]], ")"]]]], ")"]]]], "\[And]", RowBox[List["x", ">", "1"]]]]]]]]

 MathML Form

 q + 1 F ~ q ( a 1 , , a p 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]] ; b 1 , , b q 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]] ; z TagBox["z", HypergeometricPFQ, Rule[Editable, True]] ) π q k = 1 q + 1 Γ ( a k ) k = 1 q + 1 u = 0 ( - x ) - u Γ ( u + a k ) u ! j = 1 q Γ ( b j - a k ) ( j = 1 j k q + 1 csc ( π ( a j - a k ) ) ) - π a k ( 1 + 2 Floor [ arg [ x - z ] 2 π ] ) x - a k q + 1 F ~ q ( u + 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 q + 1 + a k ; 1 x ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["q", "+", "1"]], TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["q", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["u", "+", 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]], HypergeometricPFQ, Rule[Editable, True]], ";", 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", RowBox[List["q", "+", "1"]]], "+", SubscriptBox["a", "k"]]], HypergeometricPFQ, Rule[Editable, True]]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[FractionBox["1", "x"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] ( z - x ) u /; { j , k } , { j , k } TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] j k 1 j q + 1 1 k q + 1 ( a j - a k TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] ) x > 1 FormBox RowBox RowBox RowBox RowBox SubscriptBox ErrorBox FormBox RowBox q + 1 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 ) RowBox FractionBox SuperscriptBox π q RowBox UnderoverscriptBox RowBox k = 1 RowBox q + 1 RowBox Γ ( SubscriptBox a k ) RowBox UnderoverscriptBox RowBox k = 1 RowBox q + 1 RowBox UnderoverscriptBox RowBox u = 0 RowBox FractionBox RowBox SuperscriptBox RowBox ( RowBox - x ) RowBox - u RowBox Γ ( RowBox u + SubscriptBox a k ) RowBox RowBox u ! RowBox UnderoverscriptBox RowBox j = 1 q RowBox Γ ( RowBox SubscriptBox b j - SubscriptBox a k ) RowBox ( RowBox UnderoverscriptBox UnderscriptBox RowBox j = 1 RowBox j k RowBox q + 1 RowBox csc ( RowBox π RowBox ( RowBox SubscriptBox a j - SubscriptBox a k ) ) ) SuperscriptBox RowBox RowBox - π SubscriptBox a k RowBox ( RowBox 1 + RowBox 2 RowBox Floor [ FractionBox RowBox arg [ RowBox x - z ] RowBox 2 π ] ) SuperscriptBox x RowBox - SubscriptBox a k TagBox TagBox RowBox RowBox SubscriptBox FormBox RowBox q + 1 TraditionalForm SubscriptBox OverscriptBox F ~ FormBox q TraditionalForm RowBox ( RowBox TagBox TagBox TagBox RowBox TagBox RowBox u + SubscriptBox a k HypergeometricPFQ Rule Editable , TagBox RowBox SubscriptBox a k - SubscriptBox b 1 + 1 HypergeometricPFQ Rule Editable , TagBox HypergeometricPFQ Rule Editable , TagBox RowBox SubscriptBox a k - SubscriptBox b q + 1 HypergeometricPFQ Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable HypergeometricPFQ Rule Editable ; TagBox RowBox TagBox RowBox 1 - SubscriptBox a 1 + SubscriptBox a k HypergeometricPFQ Rule Editable , TagBox HypergeometricPFQ Rule Editable , TagBox RowBox 1 - SubscriptBox a RowBox k - 1 + SubscriptBox a k HypergeometricPFQ Rule Editable , TagBox RowBox 1 - SubscriptBox a RowBox k + 1 + SubscriptBox a k HypergeometricPFQ Rule Editable , TagBox HypergeometricPFQ Rule Editable , TagBox RowBox 1 - SubscriptBox a RowBox q + 1 + SubscriptBox a k HypergeometricPFQ Rule Editable HypergeometricPFQ Rule Editable ; TagBox FractionBox 1 x HypergeometricPFQ Rule Editable ) InterpretTemplate Function HypergeometricPFQ Slot 1 Slot 2 Slot 3 Rule Editable HypergeometricPFQ SuperscriptBox RowBox ( RowBox z - x ) u /; RowBox RowBox SubscriptBox RowBox RowBox { RowBox j , k } , RowBox RowBox RowBox { RowBox j , k } TagBox Function RowBox j k RowBox 1 j RowBox q + 1 RowBox 1 k RowBox q + 1 RowBox ( RowBox RowBox SubscriptBox a j - SubscriptBox a k TagBox Function ) RowBox x > 1 TraditionalForm [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a_", "1"], ",", "\[Ellipsis]_", ",", SubscriptBox["a_", RowBox[List["q_", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b_", "1"], ",", "\[Ellipsis]_", ",", SubscriptBox["b_", "q_"]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[Pi]", "q"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["q", "+", "1"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["u", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "x"]], ")"]], RowBox[List["-", "u"]]], " ", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["a", "k"], "+", "u"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], RowBox[List["q", "+", "1"]]], RowBox[List["If", "[", RowBox[List[RowBox[List["j", "\[Equal]", "k"]], ",", "1", ",", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[SubscriptBox["a", "j"], "-", SubscriptBox["a", "k"]]], ")"]]]], "]"]]]], "]"]]]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", SubscriptBox["a", "k"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["x", "-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]], ")"]]]]], " ", SuperscriptBox["x", RowBox[List["-", SubscriptBox["a", "k"]]]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["a", "k"], "+", "u"]], ",", RowBox[List["1", "+", SubscriptBox["a", "k"], "-", SubscriptBox["bb", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["a", "k"], "-", SubscriptBox["bb", "q"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "k"], "-", SubscriptBox["aa", "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", RowBox[List["q", "+", "1"]]]]]]], "}"]], ",", FractionBox["1", "x"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "u"]]], RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "q"], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "j"], "-", SubscriptBox["a", "k"]]], "]"]]]], ")"]], " ", RowBox[List["u", "!"]]]]]]]]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], RowBox[List["q", "+", "1"]]], RowBox[List["Gamma", "[", SubscriptBox["a", "k"], "]"]]]]], "/;", RowBox[List[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]", RowBox[List["q", "+", "1"]]]], "&&", RowBox[List["1", "\[LessEqual]", "k", "\[LessEqual]", RowBox[List["q", "+", "1"]]]]]]]]], RowBox[List["(", RowBox[List["!", RowBox[List[RowBox[List[SubscriptBox["a", "j"], "-", SubscriptBox["a", "k"]]], "\[Element]", "Integers"]]]], ")"]]]], "&&", RowBox[List["x", ">", "1"]]]]]]]]]]

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

 2007-05-02