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 HypergeometricPFQRegularized

 http://functions.wolfram.com/07.32.06.0020.01

 Input Form

 HypergeometricPFQRegularized[{Subscript[a, 1], \[Ellipsis], Subscript[a, q + 1]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, z] == AsymptoticHypergeometricPFQRegularizedSeries[Power][ {Subscript[a, 1], \[Ellipsis], Subscript[a, p]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, {z, Infinity, Infinity}] /; !IntervalMemberQ[Interval[{0, 1}], z] && Element[Subscript[a, k] - Subscript[a, k - 1], Integers] && Subscript[a, k] - Subscript[a, k - 1] >= 0 && 2 <= k <= r && !Element[Subscript[a, k] - Subscript[a, 1], Integers] && r + 1 <= k <= q + 1 && ForAll[{j, k}, Element[{j, k}, Integers] && j != k && r + 1 <= j <= q + 1 && r + 1 <= k <= q + 1, !Element[Subscript[a, j] - Subscript[a, k], Integers]] && Element[r, {2, 3, 4}]

 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[RowBox[List["AsymptoticHypergeometricPFQRegularizedSeries", "[", "Power", "]"]], "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]], ",", RowBox[List["{", RowBox[List["z", ",", "\[Infinity]", ",", "\[Infinity]"]], "}"]]]], "]"]]]], "/;", "\[InvisibleSpace]", RowBox[List[RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["0", ",", "1"]], "}"]], "]"]], ",", "z"]], "]"]], "]"]], "\[And]", RowBox[List[RowBox[List[SubscriptBox["a", "k"], "-", SubscriptBox["a", RowBox[List["k", "-", "1"]]]]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List[SubscriptBox["a", "k"], "-", SubscriptBox["a", RowBox[List["k", "-", "1"]]]]], "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["2", "\[LessEqual]", "k", "\[LessEqual]", "r"]], "\[And]", RowBox[List["Not", "[", RowBox[List[RowBox[List[SubscriptBox["a", "k"], "-", SubscriptBox["a", "1"]]], "\[Element]", "Integers"]], "]"]], "\[And]", RowBox[List[RowBox[List["r", "+", "1"]], "\[LessEqual]", "k", "\[LessEqual]", RowBox[List["q", "+", "1"]]]], "\[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[RowBox[List["r", "+", "1"]], "\[LessEqual]", "j", "\[LessEqual]", RowBox[List["q", "+", "1"]]]], "\[And]", RowBox[List[RowBox[List["r", "+", "1"]], "\[LessEqual]", "k", "\[LessEqual]", RowBox[List["q", "+", "1"]]]]]]]]], RowBox[List["(", RowBox[List["Not", "[", RowBox[List[RowBox[List[SubscriptBox["a", "j"], "-", SubscriptBox["a", "k"]]], "\[Element]", "Integers"]], "]"]], ")"]]]], "\[And]", RowBox[List["r", "\[Element]", RowBox[List["{", RowBox[List["2", ",", "3", ",", "4"]], "}"]]]]]]]]]]

 MathML Form

 q + 1 F ~ q ( a 1 , , a q + 1 ; b 1 , , b q ; z ) 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[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", RowBox[List["q", "+", "1"]]], 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] 𝒜 F ~ ( power ) ( a 1 , , a p ; b 1 , , b q ; { z , , } ) /; z ( 0 , 1 ) a k - a k - 1 TagBox["\[DoubleStruckCapitalN]", Function[Integers]] 2 k r a k - a 1 r + 1 k q + 1 TagBox[RowBox[List["\[DoubleStruckCapitalZ]", "\[And]", RowBox[List[RowBox[List["r", "+", "1"]], "\[LessEqual]", "k", "\[LessEqual]", RowBox[List["q", "+", "1"]]]]]], Function[Integers]] { j , k } , { j , k } TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] j k r + 1 j q + 1 r + 1 k q + 1 ( a j - a k TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] ) r { 2 , 3 , 4 } FormBox RowBox RowBox TagBox TagBox RowBox RowBox SubscriptBox 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 RowBox q + 1 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 SubsuperscriptBox 𝒜 OverscriptBox F ~ RowBox ( power ) [ RowBox GridBox ErrorBox RowBox SubscriptBox a 1 , , RowBox SubscriptBox a p ; RowBox SubscriptBox b 1 , , RowBox SubscriptBox b q ; RowBox { RowBox z , , } ] /; RowBox RowBox z RowBox ( RowBox 0 , 1 ) RowBox RowBox SubscriptBox a k - SubscriptBox a RowBox k - 1 TagBox Function RowBox 2 k r RowBox RowBox SubscriptBox a k - SubscriptBox a 1 TagBox RowBox RowBox RowBox r + 1 k RowBox q + 1 Function RowBox SubscriptBox RowBox RowBox { RowBox j , k } , RowBox RowBox RowBox { RowBox j , k } TagBox Function RowBox j k RowBox RowBox r + 1 j RowBox q + 1 RowBox RowBox r + 1 k RowBox q + 1 RowBox ( RowBox RowBox SubscriptBox a j - SubscriptBox a k TagBox Function ) RowBox r RowBox { RowBox 2 , 3 , 4 } 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[RowBox[List[RowBox[List["AsymptoticHypergeometricPFQRegularizedSeries", "[", "Power", "]"]], "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["aa", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["aa", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["bb", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["bb", "q"]]], "}"]], ",", RowBox[List["{", RowBox[List["z", ",", "\[Infinity]", ",", "\[Infinity]"]], "}"]]]], "]"]], "/;", RowBox[List[RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["0", ",", "1"]], "}"]], "]"]], ",", "z"]], "]"]]]], "&&", RowBox[List[RowBox[List[SubscriptBox["a", "k"], "-", SubscriptBox["a", RowBox[List["k", "-", "1"]]]]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[SubscriptBox["a", "k"], "-", SubscriptBox["a", RowBox[List["k", "-", "1"]]]]], "\[GreaterEqual]", "0"]], "&&", RowBox[List["2", "\[LessEqual]", "k", "\[LessEqual]", "r"]], "&&", RowBox[List["!", RowBox[List[RowBox[List[SubscriptBox["a", "k"], "-", SubscriptBox["aa", "1"]]], "\[Element]", "Integers"]]]], "&&", RowBox[List[RowBox[List["r", "+", "1"]], "\[LessEqual]", "k", "\[LessEqual]", RowBox[List["q", "+", "1"]]]], "&&", 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[RowBox[List["r", "+", "1"]], "\[LessEqual]", "j", "\[LessEqual]", RowBox[List["q", "+", "1"]]]], "&&", RowBox[List[RowBox[List["r", "+", "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["r", "\[Element]", RowBox[List["{", RowBox[List["2", ",", "3", ",", "4"]], "}"]]]]]]]]]]]]

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

 2001-10-29