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 HypergeometricPFQ

 http://functions.wolfram.com/07.27.03.0002.01

 Input Form

 HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3]}, {Subscript[b, 1], Subscript[b, 2]}, 1] == ((Gamma[Subscript[\[Psi], 3]] Product[Gamma[Subscript[b, k]], {k, 1, 2}])/ Product[Gamma[Subscript[a, k]], {k, 3, 3}]) Sum[(Pochhammer[Subscript[\[Psi], 3], k] HypergeometricPFQExpansionCoefficient[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3]}, {Subscript[b, 1], Subscript[b, 2]}, k])/(Gamma[Subscript[\[Psi], 3] + Subscript[a, 1] + k] Gamma[Subscript[\[Psi], 3] + Subscript[a, 2] + k]), {k, 0, Infinity}] /; Subscript[\[Psi], 3] == Subscript[b, 1] + Subscript[b, 2] - Subscript[a, 1] - Subscript[a, 2] - Subscript[a, 3] && Re[Subscript[\[Psi], 3]] > 0 && Re[Subscript[a, 3]] > 0

 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"]]], "}"]], ",", "1"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["\[Psi]", "3"], "]"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "2"], RowBox[List["Gamma", "[", SubscriptBox["b", "k"], "]"]]]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "3"]], "3"], RowBox[List["Gamma", "[", SubscriptBox["a", "k"], "]"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["\[Psi]", "3"], ",", "k"]], "]"]], RowBox[List["HypergeometricPFQExpansionCoefficient", "[", 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"]]], "}"]], ",", "k"]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["\[Psi]", "3"], "+", SubscriptBox["a", "1"], "+", "k"]], "]"]], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["\[Psi]", "3"], "+", SubscriptBox["a", "2"], "+", "k"]], "]"]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["\[Psi]", "3"], "\[Equal]", RowBox[List[SubscriptBox["b", "1"], "+", SubscriptBox["b", "2"], "-", SubscriptBox["a", "1"], "-", SubscriptBox["a", "2"], "-", SubscriptBox["a", "3"]]]]], "\[And]", RowBox[List[RowBox[List["Re", "[", SubscriptBox["\[Psi]", "3"], "]"]], ">", "0"]], "\[And]", RowBox[List[RowBox[List["Re", "[", SubscriptBox["a", "3"], "]"]], ">", "0"]]]]]]]]

 MathML Form

 3 F 2 ( a 1 , a 2 , a 3 ; b 1 , b 2 ; 1 ) 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["1", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] Γ ( ψ 3 ) k = 1 2 Γ ( b k ) k = 3 3 Γ ( a k ) k = 0 ( ψ 3 ) k k ( 3 ) ( { a 1 , a 2 , a 3 } , { b 1 , b 2 } ) Γ ( k + a 1 + ψ 3 ) Γ ( k + a 2 + ψ 3 ) TagBox[TagBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["\[Psi]", "3"], ")"]], "k"], Pochhammer], " ", RowBox[List[SubsuperscriptBox["\[ScriptCapitalE]", "k", RowBox[List["(", "3", ")"]]], "(", 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"]]], "}"]]]], ")"]]]], RowBox[List[RowBox[List["\[CapitalGamma]", "(", RowBox[List["k", "+", SubscriptBox["a", "1"], "+", SubscriptBox["\[Psi]", "3"]]], ")"]], " ", RowBox[List["\[CapitalGamma]", "(", RowBox[List["k", "+", SubscriptBox["a", "2"], "+", SubscriptBox["\[Psi]", "3"]]], ")"]]]]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] /; ψ 3 b 1 + b 2 - a 1 - a 2 - a 3 Re ( ψ 3 ) > 0 Re ( a 3 ) > 0 3 F 2 ( a 1 , a 2 , a 3 ; b 1 , b 2 ; 1 ) 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["1", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] Γ ( ψ 3 ) k = 1 2 Γ ( b k ) k = 3 3 Γ ( a k ) k = 0 ( ψ 3 ) k k ( 3 ) ( { a 1 , a 2 , a 3 } , { b 1 , b 2 } ) Γ ( k + a 1 + ψ 3 ) Γ ( k + a 2 + ψ 3 ) TagBox[TagBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["\[Psi]", "3"], ")"]], "k"], Pochhammer], " ", RowBox[List[SubsuperscriptBox["\[ScriptCapitalE]", "k", RowBox[List["(", "3", ")"]]], "(", 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"]]], "}"]]]], ")"]]]], RowBox[List[RowBox[List["\[CapitalGamma]", "(", RowBox[List["k", "+", SubscriptBox["a", "1"], "+", SubscriptBox["\[Psi]", "3"]]], ")"]], " ", RowBox[List["\[CapitalGamma]", "(", RowBox[List["k", "+", SubscriptBox["a", "2"], "+", SubscriptBox["\[Psi]", "3"]]], ")"]]]]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] /; ψ 3 b 1 + b 2 - a 1 - a 2 - a 3 Re ( ψ 3 ) > 0 Re ( a 3 ) > 0 [/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"]]], "}"]], ",", "1"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["\[Psi]", "3"], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "2"], RowBox[List["Gamma", "[", SubscriptBox["b", "k"], "]"]]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["\[Psi]", "3"], ",", "k"]], "]"]], " ", RowBox[List["HypergeometricPFQExpansionCoefficient", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["aa", "1"], ",", SubscriptBox["aa", "2"], ",", SubscriptBox["aa", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["bb", "1"], ",", SubscriptBox["bb", "2"]]], "}"]], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["\[Psi]", "3"], "+", SubscriptBox["aa", "1"], "+", "k"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["\[Psi]", "3"], "+", SubscriptBox["aa", "2"], "+", "k"]], "]"]]]]]]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "3"]], "3"], RowBox[List["Gamma", "[", SubscriptBox["a", "k"], "]"]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["\[Psi]", "3"], "\[Equal]", RowBox[List[SubscriptBox["bb", "1"], "+", SubscriptBox["bb", "2"], "-", SubscriptBox["aa", "1"], "-", SubscriptBox["aa", "2"], "-", SubscriptBox["aa", "3"]]]]], "&&", RowBox[List[RowBox[List["Re", "[", SubscriptBox["\[Psi]", "3"], "]"]], ">", "0"]], "&&", RowBox[List[RowBox[List["Re", "[", SubscriptBox["aa", "3"], "]"]], ">", "0"]]]]]]]]]]

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

 2001-10-29