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 HypergeometricPFQRegularized

 http://functions.wolfram.com/07.32.06.0007.01

 Input Form

 HypergeometricPFQRegularized[{Subscript[a, 1], \[Ellipsis], Subscript[a, q + 1]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, z] == (1/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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 MathML Form

 q + 1 F ~ q ( a 1 , , a q + 1 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]] ; b 1 , , b q 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]] ; z TagBox["z", HypergeometricPFQ, Rule[Editable, True]] ) 1 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]] FormBox RowBox RowBox RowBox RowBox SubscriptBox ErrorBox FormBox RowBox q + 1 TraditionalForm SubscriptBox OverscriptBox F ~ FormBox q TraditionalForm RowBox ( RowBox TagBox TagBox RowBox TagBox SubscriptBox a 1 HypergeometricPFQ Rule Editable , TagBox HypergeometricPFQ Rule Editable , TagBox SubscriptBox a RowBox q + 1 HypergeometricPFQ Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable ; TagBox TagBox RowBox TagBox SubscriptBox b 1 HypergeometricPFQ Rule Editable , TagBox HypergeometricPFQ Rule Editable , TagBox SubscriptBox b q HypergeometricPFQ Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable ; TagBox z HypergeometricPFQ Rule Editable ) RowBox FractionBox 1 RowBox UnderoverscriptBox RowBox k = 1 RowBox q + 1 RowBox Γ ( SubscriptBox a k ) RowBox ( RowBox RowBox SuperscriptBox RowBox ( RowBox z - 1 ) SubscriptBox ψ q RowBox UnderoverscriptBox RowBox j = 0 RowBox FractionBox RowBox TagBox SubscriptBox RowBox ( RowBox SubscriptBox a 1 + SubscriptBox ψ q ) j Pochhammer TagBox SubscriptBox RowBox ( RowBox SubscriptBox a 2 + SubscriptBox ψ q ) j Pochhammer RowBox RowBox j ! RowBox RowBox ( RowBox j + SubscriptBox ψ q ) ! RowBox ( RowBox RowBox UnderoverscriptBox RowBox k = RowBox j + 1 RowBox FractionBox RowBox SuperscriptBox RowBox ( RowBox - 1 ) j RowBox j ! RowBox RowBox ( RowBox k - j - 1 ) ! RowBox TagBox SubscriptBox RowBox ( RowBox SubscriptBox a 1 + SubscriptBox ψ q ) k Pochhammer TagBox SubscriptBox RowBox ( RowBox SubscriptBox a 2 + SubscriptBox ψ q ) k Pochhammer RowBox SubsuperscriptBox k RowBox ( q ) ( RowBox RowBox { RowBox SubscriptBox a 1 , , SubscriptBox a RowBox q + 1 } , RowBox { RowBox SubscriptBox b 1 , , SubscriptBox b q } ) + RowBox UnderoverscriptBox RowBox k = 0 j RowBox FractionBox TagBox SubscriptBox RowBox ( RowBox - j ) k Pochhammer RowBox TagBox SubscriptBox RowBox ( RowBox SubscriptBox a 1 + SubscriptBox ψ q ) k Pochhammer TagBox SubscriptBox RowBox ( RowBox SubscriptBox a 2 + SubscriptBox ψ q ) k Pochhammer RowBox ( RowBox RowBox - RowBox log ( RowBox 1 - z ) + RowBox TagBox ψ PolyGamma ( RowBox j - k + 1 ) + RowBox TagBox ψ PolyGamma ( RowBox j + SubscriptBox ψ q + 1 ) - RowBox TagBox ψ PolyGamma ( RowBox j + SubscriptBox a 1 + SubscriptBox ψ q ) - RowBox TagBox ψ PolyGamma ( RowBox j + SubscriptBox a 2 + SubscriptBox ψ q ) ) RowBox SubsuperscriptBox k RowBox ( q ) ( RowBox RowBox { RowBox SubscriptBox a 1 , , SubscriptBox a RowBox q + 1 } , RowBox { RowBox SubscriptBox b 1 , , SubscriptBox b q } ) ) SuperscriptBox RowBox ( RowBox 1 - z ) j + RowBox UnderoverscriptBox RowBox j = 0 RowBox SubscriptBox ψ q - 1 RowBox FractionBox RowBox RowBox Γ ( RowBox j + SubscriptBox a 1 ) RowBox Γ ( RowBox j + SubscriptBox a 2 ) RowBox j ! RowBox UnderoverscriptBox RowBox k = 0 RowBox FractionBox RowBox RowBox RowBox ( RowBox k - j + SubscriptBox ψ q - 1 ) ! RowBox SubsuperscriptBox k RowBox ( q ) ( RowBox RowBox { RowBox SubscriptBox a 1 , , SubscriptBox a RowBox q + 1 } , RowBox { RowBox SubscriptBox b 1 , , SubscriptBox b q } ) RowBox RowBox Γ ( RowBox k + SubscriptBox a 1 + SubscriptBox ψ q ) RowBox Γ ( RowBox k + SubscriptBox a 2 + SubscriptBox ψ q ) SuperscriptBox RowBox ( RowBox z - 1 ) j ) /; RowBox RowBox RowBox RowBox 1 - z < 1 RowBox SubscriptBox ψ q RowBox RowBox UnderoverscriptBox RowBox j = 1 q SubscriptBox b j - RowBox UnderoverscriptBox RowBox j = 1 RowBox q + 1 SubscriptBox a j RowBox q > 1 RowBox SubscriptBox ψ q TagBox Function TraditionalForm [/itex]

 Rule Form

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

 2001-10-29