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/07.32.06.0031.01

 Input Form

 HypergeometricPFQRegularized[{Subscript[a, 1], \[Ellipsis], Subscript[a, p]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, p + 1]}, z] \[Proportional] (1/Product[Gamma[Subscript[a, k]], {k, 1, p}]) Sum[(((Gamma[Subscript[a, k]] Product[If[j == k, 1, Gamma[Subscript[a, j] - Subscript[a, k]]], {j, 1, p}])/ Product[Gamma[Subscript[b, j] - Subscript[a, k]], {j, 1, p + 1}]) (1 + O[1/z]))/(-z)^Subscript[a, k], {k, 1, p}] + (1/Product[Gamma[Subscript[a, k]], {k, 1, p}]) (1/Sqrt[Pi]) (-z)^\[Chi] (Cos[2 Sqrt[-z] + \[Chi] Pi] (1 + O[1/z]) + (Subscript[c, 1]/(2 Sqrt[-z])) Sin[2 Sqrt[-z] + Pi \[Chi]] (1 + O[1/z])) /; (Abs[z] -> Infinity) && \[Chi] == (1/2) (1/2 + Subscript[A, p] - Subscript[B, p + 1]) && Subscript[c, 1] == 2 (\[GothicCapitalB] - \[GothicCapitalA] + (1/4) (Subscript[A, p] - Subscript[B, p + 1]) (3 Subscript[A, p] + Subscript[B, p + 1] - 2) - 3/16) && Subscript[A, p] == Sum[Subscript[a, k], {k, 1, p}] && Subscript[B, p + 1] == Sum[Subscript[b, k], {k, 1, p + 1}] && \[GothicCapitalA] == Sum[Subscript[a, s] Subscript[a, j], {s, 2, p}, {j, 1, s - 1}] && \[GothicCapitalB] == Sum[Subscript[b, s] Subscript[b, j], {s, 2, p + 1}, {j, 1, s - 1}] && 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["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", RowBox[List["p", "+", "1"]]]]], "}"]], ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[FractionBox["1", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "p"], RowBox[List["Gamma", "[", SubscriptBox["a", "k"], "]"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "p"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["a", "k"], "]"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "p"], RowBox[List["If", "[", RowBox[List[RowBox[List["j", "\[Equal]", "k"]], ",", "1", ",", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["a", "j"], "-", SubscriptBox["a", "k"]]], "]"]]]], "]"]]]]]], RowBox[List[" ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], RowBox[List["p", "+", "1"]]], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "j"], "-", SubscriptBox["a", "k"]]], "]"]]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", SubscriptBox["a", "k"]]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]]]]]], "+", RowBox[List[FractionBox["1", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "p"], RowBox[List["Gamma", "[", SubscriptBox["a", "k"], "]"]]]]], FractionBox["1", SqrtBox["\[Pi]"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], "\[Chi]"], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List[RowBox[List["2", " ", SqrtBox[RowBox[List["-", "z"]]]]], "+", RowBox[List["\[Chi]", " ", "\[Pi]"]]]], "]"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List[" ", SubscriptBox["c", "1"]]], RowBox[List["2", SqrtBox[RowBox[List["-", "z"]]]]]], RowBox[List["Sin", "[", RowBox[List[RowBox[List["2", " ", SqrtBox[RowBox[List["-", "z"]]]]], "+", RowBox[List["\[Pi]", " ", "\[Chi]"]]]], "]"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "\[And]", RowBox[List["\[Chi]", "\[Equal]", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", SubscriptBox["A", "p"], "-", SubscriptBox["B", RowBox[List["p", "+", "1"]]]]], ")"]]]]]], "\[And]", RowBox[List[SubscriptBox["c", "1"], "\[Equal]", RowBox[List["2", " ", RowBox[List["(", RowBox[List["\[GothicCapitalB]", "-", "\[GothicCapitalA]", "+", RowBox[List[FractionBox["1", "4"], RowBox[List["(", RowBox[List[SubscriptBox["A", "p"], "-", SubscriptBox["B", RowBox[List["p", "+", "1"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", SubscriptBox["A", "p"]]], "+", SubscriptBox["B", RowBox[List["p", "+", "1"]]], "-", "2"]], ")"]]]], "-", FractionBox["3", "16"]]], ")"]]]]]], "\[And]", RowBox[List[SubscriptBox["A", "p"], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "p"], SubscriptBox["a", "k"]]]]], "\[And]", RowBox[List[SubscriptBox["B", RowBox[List["p", "+", "1"]]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["p", "+", "1"]]], SubscriptBox["b", "k"]]]]], "\[And]", RowBox[List["\[GothicCapitalA]", "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "2"]], "p"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["s", "-", "1"]]], RowBox[List[SubscriptBox["a", "s"], SubscriptBox["a", "j"]]]]]]]]], "\[And]", RowBox[List["\[GothicCapitalB]", "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "2"]], RowBox[List["p", "+", "1"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["s", "-", "1"]]], RowBox[List[SubscriptBox["b", "s"], SubscriptBox["b", "j"]]]]]]]]], "\[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["(", RowBox[List["Not", "[", RowBox[List[RowBox[List[SubscriptBox["a", "j"], "-", SubscriptBox["a", "k"]]], "\[Element]", "Integers"]], "]"]], ")"]]]]]]]]]]

 MathML Form

 p F ~ p + 1 ( a 1 , , a p ; b 1 , , b p + 1 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["p", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox[RowBox[List["p", "+", "1"]], 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", RowBox[List["p", "+", "1"]]], 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] 1 π k = 1 p Γ ( a k ) ( - z ) χ ( cos ( π χ + 2 - z ) ( 1 + O ( 1 z ) ) + c 1 2 - z sin ( π χ + 2 - z ) ( 1 + O ( 1 z ) ) ) + 1 k = 1 p Γ ( a k ) k = 1 p Γ ( a k ) j = 1 j k p Γ ( a j - a k ) j = 1 p + 1 Γ ( b j - a k ) ( - z ) - a k ( 1 + O ( 1 z ) ) /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) χ 1 2 ( A p - B p + 1 + 1 2 ) c 1 2 ( 𝔅 - 𝔄 + 1 4 ( 3 A p + B p + 1 - 2 ) ( A p - B p + 1 ) - 3 16 ) A p k = 1 p a k B p + 1 k = 1 p + 1 b k 𝔄 s = 2 p j = 1 s - 1 a s a j 𝔅 s = 2 p + 1 j = 1 s - 1 b s b j { j , k } , { j , k } TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] j k 1 j p 1 k p ( a j - a k TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] ) FormBox RowBox RowBox TagBox TagBox RowBox RowBox SubscriptBox FormBox p TraditionalForm SubscriptBox OverscriptBox F ~ FormBox RowBox p + 1 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 RowBox p + 1 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 RowBox FractionBox 1 RowBox SqrtBox π RowBox UnderoverscriptBox RowBox k = 1 p RowBox Γ ( SubscriptBox a k ) SuperscriptBox RowBox ( RowBox - z ) χ RowBox ( RowBox RowBox RowBox cos ( RowBox RowBox π χ + RowBox 2 SqrtBox RowBox - z ) RowBox ( RowBox 1 + RowBox O ( FractionBox 1 z ) ) + RowBox FractionBox SubscriptBox c 1 RowBox 2 SqrtBox RowBox - z RowBox sin ( RowBox RowBox π χ + RowBox 2 SqrtBox RowBox - z ) RowBox ( RowBox 1 + RowBox O ( FractionBox 1 z ) ) ) + RowBox FractionBox 1 RowBox UnderoverscriptBox RowBox k = 1 p RowBox Γ ( SubscriptBox a k ) RowBox UnderoverscriptBox RowBox k = 1 p ErrorBox RowBox FractionBox RowBox RowBox Γ ( SubscriptBox a k ) RowBox UnderoverscriptBox UnderscriptBox RowBox j = 1 RowBox j k p RowBox Γ ( RowBox SubscriptBox a j - SubscriptBox a k ) RowBox UnderoverscriptBox RowBox j = 1 RowBox p + 1 RowBox Γ ( RowBox SubscriptBox b j - SubscriptBox a k ) SuperscriptBox RowBox ( RowBox - z ) RowBox - SubscriptBox a k RowBox ( RowBox 1 + RowBox O ( FractionBox 1 z ) ) /; RowBox RowBox ( RowBox RowBox z ) RowBox χ RowBox FractionBox 1 2 RowBox ( RowBox SubscriptBox A p - SubscriptBox B RowBox p + 1 + FractionBox 1 2 ) RowBox SubscriptBox c 1 RowBox 2 RowBox ( RowBox 𝔅 - 𝔄 + RowBox FractionBox 1 4 RowBox ( RowBox RowBox 3 SubscriptBox A p + SubscriptBox B RowBox p + 1 - 2 ) RowBox ( RowBox SubscriptBox A p - SubscriptBox B RowBox p + 1 ) - FractionBox 3 16 ) RowBox SubscriptBox A p RowBox UnderoverscriptBox RowBox k = 1 p SubscriptBox a k RowBox SubscriptBox B RowBox p + 1 RowBox UnderoverscriptBox RowBox k = 1 RowBox p + 1 SubscriptBox b k RowBox 𝔄 RowBox UnderoverscriptBox RowBox s = 2 p RowBox UnderoverscriptBox RowBox j = 1 RowBox s - 1 RowBox SubscriptBox a s SubscriptBox a j RowBox 𝔅 RowBox UnderoverscriptBox RowBox s = 2 RowBox p + 1 RowBox UnderoverscriptBox RowBox j = 1 RowBox s - 1 RowBox SubscriptBox b s SubscriptBox b j RowBox SubscriptBox RowBox RowBox { RowBox j , k } , RowBox RowBox RowBox { RowBox j , k } TagBox Function RowBox j k RowBox 1 j p RowBox 1 k p RowBox ( RowBox RowBox SubscriptBox a j - SubscriptBox a k TagBox Function ) 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_", "p_"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b_", "1"], ",", "\[Ellipsis]_", ",", SubscriptBox["b_", RowBox[List["p_", "+", "1"]]]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "p"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["a", "k"], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "p"], RowBox[List["If", "[", RowBox[List[RowBox[List["j", "\[Equal]", "k"]], ",", "1", ",", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["a", "j"], "-", SubscriptBox["a", "k"]]], "]"]]]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", SubscriptBox["a", "k"]]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], RowBox[List["p", "+", "1"]]], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "j"], "-", SubscriptBox["a", "k"]]], "]"]]]]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "p"], RowBox[List["Gamma", "[", SubscriptBox["a", "k"], "]"]]]]], "+", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], "\[Chi]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List[RowBox[List["2", " ", SqrtBox[RowBox[List["-", "z"]]]]], "+", RowBox[List["\[Chi]", " ", "\[Pi]"]]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], "+", FractionBox[RowBox[List[SubscriptBox["c", "1"], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["2", " ", SqrtBox[RowBox[List["-", "z"]]]]], "+", RowBox[List["\[Pi]", " ", "\[Chi]"]]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "z"]]]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "p"], RowBox[List["Gamma", "[", SubscriptBox["a", "k"], "]"]]]], ")"]], " ", SqrtBox["\[Pi]"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "&&", RowBox[List["\[Chi]", "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", SubscriptBox["A", "p"], "-", SubscriptBox["B", RowBox[List["p", "+", "1"]]]]], ")"]]]]]], "&&", RowBox[List[SubscriptBox["c", "1"], "\[Equal]", RowBox[List["2", " ", RowBox[List["(", RowBox[List["\[GothicCapitalB]", "-", "\[GothicCapitalA]", "+", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[SubscriptBox["A", "p"], "-", SubscriptBox["B", RowBox[List["p", "+", "1"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", SubscriptBox["A", "p"]]], "+", SubscriptBox["B", RowBox[List["p", "+", "1"]]], "-", "2"]], ")"]]]], "-", FractionBox["3", "16"]]], ")"]]]]]], "&&", RowBox[List[SubscriptBox["A", "p"], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "p"], SubscriptBox["a", "k"]]]]], "&&", RowBox[List[SubscriptBox["B", RowBox[List["p", "+", "1"]]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["p", "+", "1"]]], SubscriptBox["b", "k"]]]]], "&&", RowBox[List["\[GothicCapitalA]", "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "2"]], "p"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["s", "-", "1"]]], RowBox[List[SubscriptBox["a", "s"], " ", SubscriptBox["a", "j"]]]]]]]]], "&&", RowBox[List["\[GothicCapitalB]", "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "2"]], RowBox[List["p", "+", "1"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["s", "-", "1"]]], RowBox[List[SubscriptBox["b", "s"], " ", SubscriptBox["b", "j"]]]]]]]]], "&&", 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)

 2001-10-29