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 HypergeometricPFQ

 http://functions.wolfram.com/07.27.06.0005.01

 Input Form

 HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3]}, {Subscript[b, 1], Subscript[b, 2]}, z] == ((Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]])/ Product[Gamma[Subscript[a, k]], {k, 1, 3}]) (Sum[Subscript[g, k][0] (1 - z)^k, {k, 0, Infinity}] + (1 - z)^Subscript[\[Psi], 2] Sum[Subscript[g, k][Subscript[\[Psi], 2]] (1 - z)^k, {k, 0, Infinity}]) /; Abs[1 - z] < 1 && Subscript[g, 0][Subscript[\[Psi], 2]] == Gamma[-Subscript[\[Psi], 2]] && Subscript[g, k][r] == ((-1)^k/k!) Gamma[Subscript[a, 1] + r + k] Gamma[Subscript[a, 2] + r + k] Gamma[Subscript[\[Psi], 2] - 2 r - k] HypergeometricPFQRegularized[{Subscript[b, 1] - Subscript[a, 3], Subscript[b, 2] - Subscript[a, 3], Subscript[\[Psi], 2] - r - k}, {Subscript[a, 1] + Subscript[\[Psi], 2], Subscript[a, 2] + Subscript[\[Psi], 2]}, 1] && Subscript[\[Psi], 2] == Subscript[b, 1] + Subscript[b, 2] - Subscript[a, 1] - Subscript[a, 2] - Subscript[a, 3] && !Element[Subscript[\[Psi], 2], Integers] && 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"]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[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", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List[SubscriptBox["g", "k"], "[", "0", "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], "k"]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], SubscriptBox["\[Psi]", "2"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List[SubscriptBox["g", "k"], "[", SubscriptBox["\[Psi]", "2"], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], "k"]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List["1", "-", "z"]], "]"]], "<", "1"]], "\[And]", RowBox[List[RowBox[List[SubscriptBox["g", "0"], "[", SubscriptBox["\[Psi]", "2"], "]"]], "\[Equal]", RowBox[List["Gamma", "[", RowBox[List["-", SubscriptBox["\[Psi]", "2"]]], "]"]]]], "\[And]", RowBox[List[RowBox[List[SubscriptBox["g", "k"], "[", "r", "]"]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], RowBox[List["k", "!"]]], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["a", "1"], "+", "r", "+", "k"]], "]"]], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["a", "2"], "+", "r", "+", "k"]], "]"]], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["\[Psi]", "2"], "-", RowBox[List["2", "r"]], "-", "k"]], "]"]], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["b", "1"], "-", SubscriptBox["a", "3"]]], ",", RowBox[List[SubscriptBox["b", "2"], "-", SubscriptBox["a", "3"]]], ",", RowBox[List[SubscriptBox["\[Psi]", "2"], "-", "r", "-", "k"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["a", "1"], "+", SubscriptBox["\[Psi]", "2"]]], ",", RowBox[List[SubscriptBox["a", "2"], "+", SubscriptBox["\[Psi]", "2"]]]]], "}"]], ",", "1"]], "]"]]]]]], "\[And]", RowBox[List[SubscriptBox["\[Psi]", "2"], "\[Equal]", RowBox[List[SubscriptBox["b", "1"], "+", SubscriptBox["b", "2"], "-", SubscriptBox["a", "1"], "-", SubscriptBox["a", "2"], "-", SubscriptBox["a", "3"]]]]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List[SubscriptBox["\[Psi]", "2"], ",", "Integers"]], "]"]], "]"]], "\[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 ; 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 = 0 g k ( 0 ) ( 1 - z ) k + ( 1 - z ) ψ 2 k = 0 g k ( ψ 2 ) ( 1 - z ) k ) /; "\[LeftBracketingBar]" 1 - z "\[RightBracketingBar]" < 1 g 0 ( ψ 2 ) Γ ( - ψ 2 ) g k ( r ) ( - 1 ) k k ! Γ ( k + r + a 1 ) Γ ( k + r + a 2 ) Γ ( ψ 2 - k - 2 r ) 3 F ~ 2 ( b 1 - a 3 , b 2 - a 3 , ψ 2 - k - r ; a 1 + ψ 2 , a 2 + ψ 2 ; 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["b", "1"], "-", SubscriptBox["a", "3"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["b", "2"], "-", SubscriptBox["a", "3"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["\[Psi]", "2"], "-", "k", "-", "r"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["a", "1"], "+", SubscriptBox["\[Psi]", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "2"], "+", SubscriptBox["\[Psi]", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] ψ 2 b 1 + b 2 - a 1 - a 2 - a 3 ψ 2 TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Re ( a 3 ) > 0 Condition HypergeometricPFQ Subscript a 1 Subscript a 2 Subscript a 3 Subscript b 1 Subscript b 2 z Gamma Subscript b 1 Gamma Subscript b 2 k 1 3 Gamma Subscript a k -1 k 0 Subscript g k 0 1 -1 z k 1 -1 z Subscript ψ 2 k 0 Subscript g k Subscript ψ 2 1 -1 z k 1 -1 z 1 Subscript g 0 Subscript ψ 2 Gamma -1 Subscript ψ 2 Subscript g k r -1 k k -1 Gamma k r Subscript a 1 Gamma k r Subscript a 2 Gamma Subscript ψ 2 -1 k -1 2 r HypergeometricPFQRegularized Subscript b 1 -1 Subscript a 3 Subscript b 2 -1 Subscript a 3 Subscript ψ 2 -1 k -1 r Subscript a 1 Subscript ψ 2 Subscript a 2 Subscript ψ 2 1 Subscript ψ 2 Subscript b 1 Subscript b 2 -1 Subscript a 1 -1 Subscript a 2 -1 Subscript a 3 Subscript ψ 2 Subscript a 3 0 [/itex]

 Rule Form

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

 2001-10-29