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 HypergeometricPFQ

 http://functions.wolfram.com/07.28.06.0010.01

 Input Form

 HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3], Subscript[a, 4]}, {Subscript[b, 1], Subscript[b, 2], Subscript[b, 3]}, z] == (Product[Gamma[Subscript[b, k]], {k, 1, 3}]/ Product[Gamma[Subscript[a, k]], {k, 1, 4}]) Sum[((Pochhammer[Subscript[a, 1], j] Pochhammer[Subscript[a, 2], j])/j!^2) (Sum[((Pochhammer[-j, k] HypergeometricPFQExpansionCoefficient[ {Subscript[a, 1], Subscript[a, 2], Subscript[a, 3], Subscript[a, 4]}, {Subscript[b, 1], Subscript[b, 2], Subscript[b, 3]}, k])/(Pochhammer[Subscript[a, 1], k] Pochhammer[Subscript[a, 2], k])) (PolyGamma[1 + j - k] + PolyGamma[1 + j] - PolyGamma[Subscript[a, 1] + j] - PolyGamma[Subscript[a, 2] + j] - Log[1 - z]), {k, 0, j}] + (-1)^j j! Sum[((k - j - 1)! HypergeometricPFQExpansionCoefficient[ {Subscript[a, 1], Subscript[a, 2], Subscript[a, 3], Subscript[a, 4]}, {Subscript[b, 1], Subscript[b, 2], Subscript[b, 3]}, k])/(Pochhammer[Subscript[a, 1], k] Pochhammer[Subscript[a, 2], k]), {k, j + 1, Infinity}]) (1 - z)^j, {j, 0, Infinity}] /; Abs[1 - z] < 1 && Subscript[\[Psi], 3] == Sum[Subscript[b, j], {j, 1, 3}] - Sum[Subscript[a, j], {j, 1, 4}] && Subscript[\[Psi], 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"], ",", SubscriptBox["a", "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", SubscriptBox["b", "3"]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "3"], RowBox[List["Gamma", "[", SubscriptBox["b", "k"], "]"]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "4"], RowBox[List["Gamma", "[", SubscriptBox["a", "k"], "]"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["a", "1"], ",", "j"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["a", "2"], ",", "j"]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["j", "!"]], ")"]], "2"]], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "j"], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "j"]], ",", "k"]], "]"]], RowBox[List["HypergeometricPFQExpansionCoefficient", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", SubscriptBox["a", "3"], ",", SubscriptBox["a", "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", SubscriptBox["b", "3"]]], "}"]], ",", "k"]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["a", "1"], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["a", "2"], ",", "k"]], "]"]]]], ")"]]]], RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "j", "-", "k"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "j"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[SubscriptBox["a", "1"], "+", "j"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[SubscriptBox["a", "2"], "+", "j"]], "]"]], "-", RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]]]], ")"]]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], RowBox[List["j", "!"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["j", "+", "1"]]]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j", "-", "1"]], ")"]], "!"]], RowBox[List["HypergeometricPFQExpansionCoefficient", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", SubscriptBox["a", "3"], ",", SubscriptBox["a", "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", SubscriptBox["b", "3"]]], "}"]], ",", "k"]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["a", "1"], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["a", "2"], ",", "k"]], "]"]]]], ")"]]]]]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], "j"]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List["1", "-", "z"]], "]"]], "<", "1"]], "\[And]", RowBox[List[SubscriptBox["\[Psi]", "3"], "\[Equal]", RowBox[List[RowBox[List[StyleBox[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "3"], Rule[LimitsPositioningTokens, List["\[Sum]", "\[Product]", "\[Intersection]", "\[Union]", "\[UnionPlus]", "\[Wedge]", "\[Vee]", "lim", "max", "min", "\[CirclePlus]", "\[CircleMinus]", "\[CircleTimes]", "\[CircleDot]"]]], SubscriptBox["b", "j"]]], "-", RowBox[List[StyleBox[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "4"], Rule[LimitsPositioningTokens, List["\[Sum]", "\[Product]", "\[Intersection]", "\[Union]", "\[UnionPlus]", "\[Wedge]", "\[Vee]", "lim", "max", "min", "\[CirclePlus]", "\[CircleMinus]", "\[CircleTimes]", "\[CircleDot]"]]], SubscriptBox["a", "j"]]]]]]], "\[And]", RowBox[List[SubscriptBox["\[Psi]", "3"], "\[Equal]", "0"]]]]]]]]

 MathML Form

 4 F 3 ( a 1 , a 2 , a 3 , a 4 ; b 1 , b 2 , b 3 ; z ) 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[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "3"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "4"], 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]], ",", TagBox[SubscriptBox["b", "3"], 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] k = 1 3 Γ ( b k ) k = 1 4 Γ ( a k ) j = 0 ( a 1 ) j TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "1"], ")"]], "j"], Pochhammer] ( a 2 ) j TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "2"], ")"]], "j"], Pochhammer] j ! 2 ( k = 0 j ( - j ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "j"]], ")"]], "k"], Pochhammer] ( - log ( 1 - z ) + ψ TagBox["\[Psi]", PolyGamma] ( j + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( j + a 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( j + a 2 ) + ψ TagBox["\[Psi]", PolyGamma] ( j - k + 1 ) ) ( a 1 ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "1"], ")"]], "k"], Pochhammer] ( a 2 ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "2"], ")"]], "k"], Pochhammer] k ( 3 ) ( { a 1 , a 2 , a 3 , a 4 } , { b 1 , b 2 , b 3 } ) + ( - 1 ) j j ! k = j + 1 ( k - j - 1 ) ! ( a 1 ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "1"], ")"]], "k"], Pochhammer] ( a 2 ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "2"], ")"]], "k"], Pochhammer] k ( 3 ) ( { a 1 , a 2 , a 3 , a 4 } , { b 1 , b 2 , b 3 } ) ) ( 1 - z ) j /; "\[LeftBracketingBar]" 1 - z "\[RightBracketingBar]" < 1 ψ 3 j = 1 3 b j - j = 1 4 a j ψ 3 0 Condition HypergeometricPFQ Subscript a 1 Subscript a 2 Subscript a 3 Subscript a 4 Subscript b 1 Subscript b 2 Subscript b 3 z k 1 3 Gamma Subscript b k k 1 4 Gamma Subscript a k -1 j 0 Pochhammer Subscript a 1 j Pochhammer Subscript a 2 j j 2 -1 k 0 j Pochhammer -1 j k -1 1 -1 z PolyGamma j 1 -1 PolyGamma j Subscript a 1 -1 PolyGamma j Subscript a 2 PolyGamma j -1 k 1 Pochhammer Subscript a 1 k Pochhammer Subscript a 2 k -1 Subscript k 3 Subscript a 1 Subscript a 2 Subscript a 3 Subscript a 4 Subscript b 1 Subscript b 2 Subscript b 3 -1 j j k j 1 k -1 j -1 Pochhammer Subscript a 1 k Pochhammer Subscript a 2 k -1 Subscript k 3 Subscript a 1 Subscript a 2 Subscript a 3 Subscript a 4 Subscript b 1 Subscript b 2 Subscript b 3 1 -1 z j 1 -1 z 1 Subscript ψ 3 j 1 3 Subscript b j -1 j 1 4 Subscript a j Subscript ψ 3 0 [/itex]

 Rule Form

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

 2001-10-29