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 HypergeometricPFQ

 http://functions.wolfram.com/07.27.06.0016.01

 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]])/ (Gamma[Subscript[a, 1]] Gamma[Subscript[a, 2]] Gamma[Subscript[a, 3]])) (-Log[1 - z] - 2 EulerGamma - PolyGamma[Subscript[a, 1]] - PolyGamma[Subscript[a, 2]] + (((Subscript[b, 1] - Subscript[a, 3]) (Subscript[b, 2] - Subscript[a, 3]))/(Subscript[a, 1] Subscript[a, 2])) HypergeometricPFQ[{Subscript[b, 1] - Subscript[a, 3] + 1, Subscript[b, 2] - Subscript[a, 3] + 1, 1, 1}, {Subscript[a, 1] + 1, Subscript[a, 2] + 1, 2}, 1]) (1 + O[z - 1]) /; (z -> 1) && Subscript[\[Psi], 2] == Subscript[b, 1] + Subscript[b, 2] - Subscript[a, 1] - Subscript[a, 2] - Subscript[a, 3] && Subscript[\[Psi], 2] == 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"]], "]"]], "\[Proportional]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["b", "1"], "]"]], RowBox[List["Gamma", "[", SubscriptBox["b", "2"], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["a", "1"], "]"]], RowBox[List["Gamma", "[", SubscriptBox["a", "2"], "]"]], RowBox[List["Gamma", "[", SubscriptBox["a", "3"], "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]]]], "-", RowBox[List["2", " ", "EulerGamma"]], "-", RowBox[List["PolyGamma", "[", SubscriptBox["a", "1"], "]"]], "-", RowBox[List["PolyGamma", "[", SubscriptBox["a", "2"], "]"]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["b", "1"], "-", SubscriptBox["a", "3"]]], ")"]], RowBox[List["(", RowBox[List[SubscriptBox["b", "2"], "-", SubscriptBox["a", "3"]]], ")"]]]], RowBox[List[SubscriptBox["a", "1"], SubscriptBox["a", "2"]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["b", "1"], "-", SubscriptBox["a", "3"], "+", "1"]], ",", RowBox[List[SubscriptBox["b", "2"], "-", SubscriptBox["a", "3"], "+", "1"]], ",", "1", ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["a", "1"], "+", "1"]], ",", RowBox[List[SubscriptBox["a", "2"], "+", "1"]], ",", "2"]], "}"]], ",", "1"]], "]"]]]]]], ")"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", RowBox[List["z", "-", "1"]], "]"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", "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[SubscriptBox["\[Psi]", "2"], "\[Equal]", "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 ) Γ ( a 1 ) Γ ( a 2 ) Γ ( a 3 ) ( - log ( 1 - z ) - ψ TagBox["\[Psi]", PolyGamma] ( a 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( a 2 ) - 2 TagBox["\[DoubledGamma]", Function[EulerGamma]] + ( b 1 - a 3 ) ( b 2 - a 3 ) a 1 a 2 4 F 3 ( - a 3 + b 1 + 1 , - a 3 + b 2 + 1 , 1 , 1 ; a 1 + 1 , a 2 + 1 , 2 ; 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["4", TraditionalForm]], SubscriptBox["F", FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[RowBox[List["-", SubscriptBox["a", "3"]]], "+", SubscriptBox["b", "1"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", SubscriptBox["a", "3"]]], "+", SubscriptBox["b", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["1", HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["a", "1"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["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] ) ( 1 + O ( z - 1 ) ) /; ( z "\[Rule]" 1 ) ψ 2 b 1 + b 2 - a 1 - a 2 - a 3 ψ 2 0 Condition Proportional 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 Gamma Subscript a 1 Gamma Subscript a 2 Gamma Subscript a 3 -1 -1 1 -1 z -1 PolyGamma Subscript a 1 -1 PolyGamma Subscript a 2 -1 2 Subscript b 1 -1 Subscript a 3 Subscript b 2 -1 Subscript a 3 Subscript a 1 Subscript a 2 -1 HypergeometricPFQ -1 Subscript a 3 Subscript b 1 1 -1 Subscript a 3 Subscript b 2 1 1 1 Subscript a 1 1 Subscript a 2 1 2 1 1 O z -1 Rule z 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 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"]]], "}"]], ",", "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["-", RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]]]], "-", RowBox[List["2", " ", "EulerGamma"]], "-", RowBox[List["PolyGamma", "[", SubscriptBox["aa", "1"], "]"]], "-", RowBox[List["PolyGamma", "[", SubscriptBox["aa", "2"], "]"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["bb", "1"], "-", SubscriptBox["aa", "3"]]], ")"]], " ", RowBox[List["(", RowBox[List[SubscriptBox["bb", "2"], "-", SubscriptBox["aa", "3"]]], ")"]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["bb", "1"], "-", SubscriptBox["aa", "3"], "+", "1"]], ",", RowBox[List[SubscriptBox["bb", "2"], "-", SubscriptBox["aa", "3"], "+", "1"]], ",", "1", ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["aa", "1"], "+", "1"]], ",", RowBox[List[SubscriptBox["aa", "2"], "+", "1"]], ",", "2"]], "}"]], ",", "1"]], "]"]]]], RowBox[List[SubscriptBox["aa", "1"], " ", SubscriptBox["aa", "2"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", RowBox[List["z", "-", "1"]], "]"]]]], ")"]]]], RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["aa", "1"], "]"]], " ", RowBox[List["Gamma", "[", SubscriptBox["aa", "2"], "]"]], " ", RowBox[List["Gamma", "[", SubscriptBox["aa", "3"], "]"]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", "1"]], ")"]], "&&", RowBox[List[SubscriptBox["\[Psi]", "2"], "\[Equal]", RowBox[List[SubscriptBox["bb", "1"], "+", SubscriptBox["bb", "2"], "-", SubscriptBox["aa", "1"], "-", SubscriptBox["aa", "2"], "-", SubscriptBox["aa", "3"]]]]], "&&", RowBox[List[SubscriptBox["\[Psi]", "2"], "\[Equal]", "0"]]]]]]]]]]

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

 2001-10-29