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 HypergeometricPFQ

 http://functions.wolfram.com/07.27.06.0025.01

 Input Form

 HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3]}, {Subscript[b, 1], Subscript[b, 2]}, z] \[Proportional] ((Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]])/ Product[Gamma[Subscript[a, k]], {k, 1, 3}]) (Sum[(((Gamma[Subscript[a, k]] Product[If[j != k, Gamma[Subscript[a, j] - Subscript[a, k]], 1], {j, 1, 3}])/ (Gamma[Subscript[b, 1] - Subscript[a, k]] Gamma[Subscript[b, 2] - Subscript[a, k]])) (1 + O[1/z]))/ (-z)^Subscript[a, k], {k, r + 1, 3}] - Sum[GammaResidue[{{0}, {Subscript[a, 1], Subscript[a, 2], Subscript[a, 3]}}, {{}, {Subscript[b, 1], Subscript[b, 2]}}, {Subscript[a, j - 1], j - 1, 0}, -z] (1 + O[1/z]), {j, 2, r + 1}]) /; (Abs[z] -> Infinity) && 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 <= 3 && ForAll[{j, k}, Element[{j, k}, Integers] && j != k && r + 1 <= j <= 3 && r + 1 <= k <= 3, !Element[Subscript[a, j] - Subscript[a, k], Integers]] && Element[r, {2, 3}]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", SubscriptBox["a", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"]]], "}"]], ",", "z"]], "]"]], "\[Proportional]", RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["b", "1"], "]"]], RowBox[List["Gamma", "[", SubscriptBox["b", "2"], "]"]]]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "3"], RowBox[List["Gamma", "[", SubscriptBox["a", "k"], "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["r", "+", "1"]]]], "3"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["a", "k"], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "3"], RowBox[List["If", "[", RowBox[List[RowBox[List["j", "\[NotEqual]", "k"]], ",", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["a", "j"], "-", SubscriptBox["a", "k"]]], "]"]], " ", ",", "1"]], "]"]]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "1"], "-", SubscriptBox["a", "k"]]], "]"]], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "2"], "-", 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["\[Sum]", RowBox[List["j", "=", "2"]], RowBox[List["r", "+", "1"]]], RowBox[List[RowBox[List["GammaResidue", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "0", "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", SubscriptBox["a", "3"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["a", RowBox[List["j", "-", "1"]]], ",", RowBox[List["j", "-", "1"]], ",", "0"]], "}"]], ",", RowBox[List["-", "z"]]]], "]"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]]]]]], ")"]]]]]], "/;", "\[InvisibleSpace]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "\[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]", "3"]], "\[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]", "3"]], "\[And]", RowBox[List[RowBox[List["r", "+", "1"]], "\[LessEqual]", "k", "\[LessEqual]", "3"]]]]]]], 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"]], "}"]]]]]]]]]]

 MathML Form

 3 F 2 ( a 1 , a 2 , a 3 ; b 1 , b 2 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "3"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["b", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "2"], 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, False]], HypergeometricPFQ] Γ ( b 1 ) Γ ( b 2 ) k = 1 3 Γ ( a k ) ( k = r + 1 3 Γ ( a k ) j = 1 j k 3 Γ ( a j - a k ) Γ ( b 1 - a k ) Γ ( b 2 - a k ) ( - z ) - a k ( 1 + O ( 1 z ) ) - j = 2 r + 1 ΓRes ( 0 ; a 1 , a 2 , a 3 ; ; b 1 , b 2 ; a j - 1 , j - 1 , 0 ; - z ) ( 1 + O ( 1 z ) ) ) /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) a k - a k - 1 TagBox["\[DoubleStruckCapitalN]", Function[Integers]] 2 k r a k - a 1 r + 1 k 3 TagBox[RowBox[List["\[DoubleStruckCapitalZ]", "\[And]", RowBox[List[RowBox[List["r", "+", "1"]], "\[LessEqual]", "k", "\[LessEqual]", "3"]]]], Function[Integers]] { j , k } , { j , k } TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] j k r + 1 j 3 r + 1 k 3 ( a j - a k TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] ) r { 2 , 3 } FormBox RowBox RowBox TagBox TagBox RowBox RowBox SubscriptBox FormBox 3 TraditionalForm SubscriptBox F FormBox 2 TraditionalForm RowBox ( RowBox TagBox TagBox RowBox TagBox SubscriptBox a 1 HypergeometricPFQ Rule Editable , TagBox SubscriptBox a 2 HypergeometricPFQ Rule Editable , TagBox SubscriptBox a 3 HypergeometricPFQ Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable ; TagBox TagBox RowBox TagBox SubscriptBox b 1 HypergeometricPFQ Rule Editable , TagBox SubscriptBox b 2 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 FractionBox RowBox RowBox Γ ( SubscriptBox b 1 ) RowBox Γ ( SubscriptBox b 2 ) RowBox UnderoverscriptBox RowBox k = 1 3 RowBox Γ ( SubscriptBox a k ) RowBox ( RowBox RowBox UnderoverscriptBox RowBox k = RowBox r + 1 3 ErrorBox RowBox FractionBox RowBox RowBox Γ ( SubscriptBox a k ) RowBox UnderoverscriptBox UnderscriptBox RowBox j = 1 RowBox j k 3 RowBox Γ ( RowBox SubscriptBox a j - SubscriptBox a k ) RowBox RowBox Γ ( RowBox SubscriptBox b 1 - SubscriptBox a k ) RowBox Γ ( RowBox SubscriptBox b 2 - SubscriptBox a k ) SuperscriptBox RowBox ( RowBox - z ) RowBox - SubscriptBox a k RowBox ( RowBox 1 + RowBox O ( FractionBox 1 z ) ) - RowBox UnderoverscriptBox RowBox j = 2 RowBox r + 1 RowBox RowBox ΓRes ( RowBox RowBox GridBox RowBox 0 ; RowBox SubscriptBox a 1 , SubscriptBox a 2 , RowBox SubscriptBox a 3 ; ; RowBox SubscriptBox b 1 , RowBox SubscriptBox b 2 ; SubscriptBox a RowBox j - 1 , RowBox j - 1 , RowBox 0 ; RowBox - z ) RowBox ( RowBox 1 + RowBox O ( FractionBox 1 z ) ) ) /; RowBox RowBox ( RowBox RowBox z ) 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 3 Function RowBox SubscriptBox RowBox RowBox { RowBox j , k } , RowBox RowBox RowBox { RowBox j , k } TagBox Function RowBox j k RowBox RowBox r + 1 j 3 RowBox RowBox r + 1 k 3 RowBox ( RowBox RowBox SubscriptBox a j - SubscriptBox a k TagBox Function ) RowBox r RowBox { RowBox 2 , 3 } TraditionalForm [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a_", "1"], ",", SubscriptBox["a_", "2"], ",", SubscriptBox["a_", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b_", "1"], ",", SubscriptBox["b_", "2"]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["bb", "1"], "]"]], " ", RowBox[List["Gamma", "[", SubscriptBox["bb", "2"], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["r", "+", "1"]]]], "3"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["a", "k"], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "3"], RowBox[List["If", "[", RowBox[List[RowBox[List["j", "\[NotEqual]", "k"]], ",", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["aa", "j"], "-", SubscriptBox["a", "k"]]], "]"]], ",", "1"]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", SubscriptBox["a", "k"]]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["bb", "1"], "-", SubscriptBox["a", "k"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["bb", "2"], "-", SubscriptBox["a", "k"]]], "]"]]]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "2"]], RowBox[List["r", "+", "1"]]], RowBox[List[RowBox[List["GammaResidue", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "0", "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["aa", "1"], ",", SubscriptBox["aa", "2"], ",", SubscriptBox["aa", "3"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["bb", "1"], ",", SubscriptBox["bb", "2"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["a", RowBox[List["j", "-", "1"]]], ",", RowBox[List["j", "-", "1"]], ",", "0"]], "}"]], ",", RowBox[List["-", "z"]]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]]]]]], ")"]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "3"], RowBox[List["Gamma", "[", SubscriptBox["a", "k"], "]"]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "&&", 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]", "3"]], "&&", 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]", "3"]], "&&", RowBox[List[RowBox[List["r", "+", "1"]], "\[LessEqual]", "k", "\[LessEqual]", "3"]]]]]]], RowBox[List["(", RowBox[List["!", RowBox[List[RowBox[List[SubscriptBox["aa", "j"], "-", SubscriptBox["a", "k"]]], "\[Element]", "Integers"]]]], ")"]]]], "&&", RowBox[List["r", "\[Element]", RowBox[List["{", RowBox[List["2", ",", "3"]], "}"]]]]]]]]]]]]

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

 2001-10-29