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 HypergeometricPFQ

 http://functions.wolfram.com/07.27.06.0015.01

 Input Form

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

 Standard Form

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 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 ( ψ 2 ) k = 1 2 Γ ( b k ) k = 3 3 Γ ( a k ) k = 0 ( ψ 2 ) k k ( 2 ) ( { a 1 , a 2 , a 3 } , { b 1 , b 2 } ) Γ ( k + a 1 + ψ 2 ) Γ ( k + a 2 + ψ 2 ) ( 1 + O ( z - 1 ) ) TagBox[TagBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["\[Psi]", "2"], ")"]], "k"], Pochhammer], " ", RowBox[List[SubsuperscriptBox["\[ScriptCapitalE]", "k", RowBox[List["(", "2", ")"]]], "(", 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]", "2"]]], ")"]], " ", RowBox[List["\[CapitalGamma]", "(", RowBox[List["k", "+", SubscriptBox["a", "2"], "+", SubscriptBox["\[Psi]", "2"]]], ")"]]]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "(", RowBox[List["z", "-", "1"]], ")"]]]], ")"]]]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] + Γ ( - ψ 2 ) k = 1 2 Γ ( b k ) k = 3 3 Γ ( a k ) ( 1 - z ) ψ 2 ( 1 + O ( z - 1 ) ) /; ( z "\[Rule]" 1 ) ψ 2 b 1 + b 2 - a 1 - a 2 - a 3 Re ( ψ 2 ) > 0 Re ( a 3 ) > 0 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 ( ψ 2 ) k = 1 2 Γ ( b k ) k = 3 3 Γ ( a k ) k = 0 ( ψ 2 ) k k ( 2 ) ( { a 1 , a 2 , a 3 } , { b 1 , b 2 } ) Γ ( k + a 1 + ψ 2 ) Γ ( k + a 2 + ψ 2 ) ( 1 + O ( z - 1 ) ) TagBox[TagBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["\[Psi]", "2"], ")"]], "k"], Pochhammer], " ", RowBox[List[SubsuperscriptBox["\[ScriptCapitalE]", "k", RowBox[List["(", "2", ")"]]], "(", 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]", "2"]]], ")"]], " ", RowBox[List["\[CapitalGamma]", "(", RowBox[List["k", "+", SubscriptBox["a", "2"], "+", SubscriptBox["\[Psi]", "2"]]], ")"]]]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "(", RowBox[List["z", "-", "1"]], ")"]]]], ")"]]]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] + Γ ( - ψ 2 ) k = 1 2 Γ ( b k ) k = 3 3 Γ ( a k ) ( 1 - z ) ψ 2 ( 1 + O ( z - 1 ) ) /; ( z "\[Rule]" 1 ) ψ 2 b 1 + b 2 - a 1 - a 2 - a 3 Re ( ψ 2 ) > 0 Re ( a 3 ) > 0 [/itex]

 Rule Form

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

 2001-10-29