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 HypergeometricPFQ

 http://functions.wolfram.com/07.31.06.0007.01

 Input Form

 HypergeometricPFQ[{Subscript[a, 1], \[Ellipsis], Subscript[a, q + 1]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, z] == (Product[Gamma[Subscript[b, k]], {k, 1, q}]/Product[Gamma[Subscript[a, k]], {k, 1, q + 1}]) (Sum[((Gamma[Subscript[a, 1] + j] Gamma[Subscript[a, 2] + j])/j!) Sum[(((Subscript[\[Psi], q] + k - j - 1)! HypergeometricPFQExpansionCoefficient[{Subscript[a, 1], \[Ellipsis], Subscript[a, q + 1]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, k])/(Gamma[Subscript[a, 1] + Subscript[\[Psi], q] + k] Gamma[Subscript[a, 2] + Subscript[\[Psi], q] + k])) (z - 1)^j, {k, 0, Infinity}], {j, 0, Subscript[\[Psi], q] - 1}] + (z - 1)^Subscript[\[Psi], q] Sum[((Pochhammer[Subscript[a, 1] + Subscript[\[Psi], q], j] Pochhammer[Subscript[a, 2] + Subscript[\[Psi], q], j])/ (j! (Subscript[\[Psi], q] + j)!)) (Sum[((-1)^j j! (k - j - 1)! HypergeometricPFQExpansionCoefficient[ {Subscript[a, 1], \[Ellipsis], Subscript[a, q + 1]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, k])/ (Pochhammer[Subscript[a, 1] + Subscript[\[Psi], q], k] Pochhammer[Subscript[a, 2] + Subscript[\[Psi], q], k]), {k, j + 1, Infinity}] + Sum[((Pochhammer[-j, k] HypergeometricPFQExpansionCoefficient[ {Subscript[a, 1], \[Ellipsis], Subscript[a, q + 1]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, k])/ (Pochhammer[Subscript[a, 1] + Subscript[\[Psi], q], k] Pochhammer[Subscript[a, 2] + Subscript[\[Psi], q], k])) (-Log[1 - z] + PolyGamma[j - k + 1] + PolyGamma[ j + Subscript[\[Psi], q] + 1] - PolyGamma[Subscript[\[Psi], q] + Subscript[a, 1] + j] - PolyGamma[Subscript[\[Psi], q] + Subscript[a, 2] + j]), {k, 0, j}]) (1 - z)^j, {j, 0, Infinity}]) /; Abs[1 - z] < 1 && Subscript[\[Psi], q] == Sum[Subscript[b, j], {j, 1, q}] - Sum[Subscript[a, j], {j, 1, q + 1}] && q > 1 && Element[Subscript[\[Psi], q], Integers] && Subscript[\[Psi], q] >= 0

 Standard Form

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"1"], "+", "j"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[SubscriptBox["\[Psi]", "q"], "+", SubscriptBox["a", "2"], "+", "j"]], "]"]]]], ")"]]]]]]]], ")"]], 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]", "q"], "\[Equal]", RowBox[List[RowBox[List[StyleBox[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "q"], 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"]], RowBox[List["q", "+", "1"]]], Rule[LimitsPositioningTokens, List["\[Sum]", "\[Product]", "\[Intersection]", "\[Union]", "\[UnionPlus]", "\[Wedge]", "\[Vee]", "lim", "max", "min", "\[CirclePlus]", "\[CircleMinus]", "\[CircleTimes]", "\[CircleDot]"]]], SubscriptBox["a", "j"]]]]]]], "\[And]", RowBox[List["q", ">", "1"]], "\[And]", RowBox[List[SubscriptBox["\[Psi]", "q"], "\[Element]", "Integers"]], "\[And]", RowBox[List[SubscriptBox["\[Psi]", "q"], "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 q + 1 F q ( a 1 , , a q + 1 ; b 1 , , b q ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["q", "+", "1"]], TraditionalForm]], SubscriptBox["F", FormBox["q", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", RowBox[List["q", "+", "1"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["b", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "q"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox["z", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] k = 1 q Γ ( b k ) k = 1 q + 1 Γ ( a k ) ( ( z - 1 ) ψ q j = 0 ( a 1 + ψ q ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], "+", SubscriptBox["\[Psi]", "q"]]], ")"]], "j"], Pochhammer] ( a 2 + ψ q ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "2"], "+", SubscriptBox["\[Psi]", "q"]]], ")"]], "j"], Pochhammer] j ! ( j + ψ q ) ! ( k = j + 1 ( - 1 ) j j ! ( k - j - 1 ) ! ( a 1 + ψ q ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], "+", SubscriptBox["\[Psi]", "q"]]], ")"]], "k"], Pochhammer] ( a 2 + ψ q ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "2"], "+", SubscriptBox["\[Psi]", "q"]]], ")"]], "k"], Pochhammer] k ( q ) ( { a 1 , , a q + 1 } , { b 1 , , b q } ) + k = 0 j ( - j ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "j"]], ")"]], "k"], Pochhammer] ( a 1 + ψ q ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], "+", SubscriptBox["\[Psi]", "q"]]], ")"]], "k"], Pochhammer] ( a 2 + ψ q ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "2"], "+", SubscriptBox["\[Psi]", "q"]]], ")"]], "k"], Pochhammer] ( - log ( 1 - z ) + ψ TagBox["\[Psi]", PolyGamma] ( j - k + 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( j + ψ q + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( j + a 1 + ψ q ) - ψ TagBox["\[Psi]", PolyGamma] ( j + a 2 + ψ q ) ) k ( q ) ( { a 1 , , a q + 1 } , { b 1 , , b q } ) ) ( 1 - z ) j + j = 0 ψ q - 1 Γ ( j + a 1 ) Γ ( j + a 2 ) j ! k = 0 ( k - j + ψ q - 1 ) ! k ( q ) ( { a 1 , , a q + 1 } , { b 1 , , b q } ) Γ ( k + a 1 + ψ q ) Γ ( k + a 2 + ψ q ) ( z - 1 ) j ) /; "\[LeftBracketingBar]" 1 - z "\[RightBracketingBar]" < 1 ψ q j = 1 q b j - j = 1 q + 1 a j q > 1 ψ q TagBox["\[DoubleStruckCapitalN]", Function[Integers]] Condition HypergeometricPFQ Subscript a 1 Subscript a q 1 Subscript b 1 Subscript b q z k 1 q Gamma Subscript b k k 1 q 1 Gamma Subscript a k -1 z -1 Subscript ψ q j 0 Pochhammer Subscript a 1 Subscript ψ q j Pochhammer Subscript a 2 Subscript ψ q j j j Subscript ψ q -1 k j 1 -1 j j k -1 j -1 Pochhammer Subscript a 1 Subscript ψ q k Pochhammer Subscript a 2 Subscript ψ q k -1 Subscript k q Subscript a 1 Subscript a q 1 Subscript b 1 Subscript b q k 0 j Pochhammer -1 j k Pochhammer Subscript a 1 Subscript ψ q k Pochhammer Subscript a 2 Subscript ψ q k -1 -1 1 -1 z PolyGamma j -1 k 1 PolyGamma j Subscript ψ q 1 -1 PolyGamma j Subscript a 1 Subscript ψ q -1 PolyGamma j Subscript a 2 Subscript ψ q Subscript k q Subscript a 1 Subscript a q 1 Subscript b 1 Subscript b q 1 -1 z j j 0 Subscript ψ q -1 Gamma j Subscript a 1 Gamma j Subscript a 2 j -1 k 0 k -1 j Subscript ψ q -1 Subscript k q Subscript a 1 Subscript a q 1 Subscript b 1 Subscript b q Gamma k Subscript a 1 Subscript ψ q Gamma k Subscript a 2 Subscript ψ q -1 z -1 j 1 -1 z 1 Subscript ψ q j 1 q Subscript b j -1 j 1 q 1 Subscript a j q 1 Subscript ψ q [/itex]

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

 2001-10-29