html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 HypergeometricPFQRegularized

 http://functions.wolfram.com/07.32.06.0008.01

 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}]) ((1 - z)^Subscript[\[Psi], q] Sum[Subscript[h, j] (1 - j)^j, {j, 0, -Subscript[\[Psi], q] - 1}] + Sum[(Subscript[u, j] + Subscript[v, j] Log[1 - z]) (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 && Subscript[h, j] == (((-1)^j Pochhammer[Subscript[a, 1] + Subscript[\[Psi], q], j] Pochhammer[Subscript[a, 2] + Subscript[\[Psi], q], j] (-Subscript[\[Psi], q] - j - 1)!)/j!) Sum[(Pochhammer[-j, k]/(Pochhammer[Subscript[\[Psi], q] + Subscript[a, 1], k] Pochhammer[Subscript[\[Psi], q] + Subscript[a, 2], k])) HypergeometricPFQExpansionCoefficient[ {Subscript[a, 1], \[Ellipsis], Subscript[a, q + 1]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, k], {k, 0, j}] && (Subscript[u, j] == ((Pochhammer[Subscript[a, 1] + Subscript[\[Psi], q], j - Subscript[\[Psi], q]] Pochhammer[Subscript[a, 2] + Subscript[\[Psi], q], j - Subscript[\[Psi], q]])/ (j! (j - Subscript[\[Psi], q])!)) ((-1)^j (j - Subscript[\[Psi], q])! Sum[((Subscript[\[Psi], q] + k - j - 1)!/ (Pochhammer[Subscript[\[Psi], q] + Subscript[a, 1], k] Pochhammer[Subscript[\[Psi], q] + Subscript[a, 2], k])) HypergeometricPFQExpansionCoefficient[{Subscript[a, 1], \[Ellipsis], Subscript[a, q + 1]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, k], {k, j - Subscript[\[Psi], q] + 1, Infinity}] + (-1)^Subscript[\[Psi], q] Sum[(Pochhammer[Subscript[\[Psi], q] - j, k]/ (Pochhammer[Subscript[\[Psi], q] + Subscript[a, 1], k] Pochhammer[Subscript[\[Psi], q] + Subscript[a, 2], k])) HypergeometricPFQExpansionCoefficient[{Subscript[a, 1], \[Ellipsis], Subscript[a, q + 1]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, k] (PolyGamma[j + 1] + PolyGamma[j - Subscript[\[Psi], q] + 1 - k] - PolyGamma[Subscript[a, 1] + j] - PolyGamma[Subscript[a, 2] + j]), {k, 0, j - Subscript[\[Psi], q]}]) /; Re[Subscript[a, 3]] > -j && \[Ellipsis] && Re[Subscript[a, q + 1]] > -j) && Subscript[v, j] == (((-1)^(Subscript[\[Psi], q] - 1) Pochhammer[Subscript[a, 1] + Subscript[\[Psi], q], j - Subscript[\[Psi], q]] Pochhammer[Subscript[a, 2] + Subscript[\[Psi], q], j - Subscript[\[Psi], q]])/ (j! (j - Subscript[\[Psi], q])!)) Sum[(Pochhammer[Subscript[\[Psi], q] - j, k]/ (Pochhammer[Subscript[\[Psi], q] + Subscript[a, 1], k] Pochhammer[Subscript[\[Psi], q] + Subscript[a, 2], k])) HypergeometricPFQExpansionCoefficient[{Subscript[a, 1], \[Ellipsis], Subscript[a, q + 1]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, k], {k, 0, j - Subscript[\[Psi], q]}] && 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 ) ( ( 1 - z ) ψ q j = 0 - ψ q - 1 h j ( 1 - j ) j + j = 0 ( u j + v j log ( 1 - z ) ) ( 1 - z ) j ) /; "\[LeftBracketingBar]" 1 - z "\[RightBracketingBar]" < 1 ψ q j = 1 q b j - j = 1 q + 1 a j q > 1 h j ( - 1 ) j ( 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 - ψ q - 1 ) ! j ! k = 0 j ( - j ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "j"]], ")"]], "k"], Pochhammer] k ( q ) ( { a 1 , , a q + 1 } , { b 1 , , b q } ) ( 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] ( u j ( a 1 + ψ q ) j - ψ q TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], "+", SubscriptBox["\[Psi]", "q"]]], ")"]], RowBox[List["j", "-", SubscriptBox["\[Psi]", "q"]]]], Pochhammer] ( a 2 + ψ q ) j - ψ q TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "2"], "+", SubscriptBox["\[Psi]", "q"]]], ")"]], RowBox[List["j", "-", SubscriptBox["\[Psi]", "q"]]]], Pochhammer] j ! ( j - ψ q ) ! ( ( - 1 ) j ( j - ψ q ) ! k = j - ψ q + 1 ( k - j + ψ q - 1 ) ! k ( q ) ( { a 1 , , a q + 1 } , { b 1 , , b q } ) ( 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] + ( - 1 ) ψ q k = 0 j - ψ q ( ψ q - j ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["\[Psi]", "q"], "-", "j"]], ")"]], "k"], Pochhammer] k ( q ) ( { a 1 , , a q + 1 } , { b 1 , , b q } ) ( 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] ( 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 - ψ q + 1 ) ) ) /; Re ( a 3 ) > - j Re ( a q + 1 ) > - j ) v j ( - 1 ) ψ q - 1 ( a 1 + ψ q ) j - ψ q TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], "+", SubscriptBox["\[Psi]", "q"]]], ")"]], RowBox[List["j", "-", SubscriptBox["\[Psi]", "q"]]]], Pochhammer] ( a 2 + ψ q ) j - ψ q TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "2"], "+", SubscriptBox["\[Psi]", "q"]]], ")"]], RowBox[List["j", "-", SubscriptBox["\[Psi]", "q"]]]], Pochhammer] j ! ( j - ψ q ) ! k = 0 j - ψ q ( ψ q - j ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["\[Psi]", "q"], "-", "j"]], ")"]], "k"], Pochhammer] k ( q ) ( { a 1 , , a q + 1 } , { b 1 , , b q } ) ( 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] - ψ q 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 1 - z ) SubscriptBox ψ q RowBox UnderoverscriptBox RowBox j = 0 RowBox RowBox - SubscriptBox ψ q - 1 RowBox SubscriptBox h j SuperscriptBox RowBox ( RowBox 1 - j ) j + RowBox UnderoverscriptBox RowBox j = 0 RowBox RowBox ( RowBox SubscriptBox u j + RowBox SubscriptBox v j RowBox log ( RowBox 1 - z ) ) SuperscriptBox RowBox ( RowBox 1 - z ) 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 h j RowBox FractionBox RowBox SuperscriptBox RowBox ( RowBox - 1 ) j 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 ( RowBox RowBox - j - SubscriptBox ψ q - 1 ) ! RowBox j ! RowBox UnderoverscriptBox RowBox k = 0 j FractionBox RowBox TagBox SubscriptBox RowBox ( RowBox - j ) 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 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 SubscriptBox u j RowBox FractionBox RowBox TagBox SubscriptBox RowBox ( RowBox SubscriptBox a 1 + SubscriptBox ψ q ) RowBox j - SubscriptBox ψ q Pochhammer TagBox SubscriptBox RowBox ( RowBox SubscriptBox a 2 + SubscriptBox ψ q ) RowBox j - SubscriptBox ψ q Pochhammer RowBox RowBox j ! RowBox RowBox ( RowBox j - SubscriptBox ψ q ) ! RowBox ( RowBox RowBox SuperscriptBox RowBox ( RowBox - 1 ) j RowBox RowBox ( RowBox j - SubscriptBox ψ q ) ! RowBox UnderoverscriptBox RowBox k = RowBox j - SubscriptBox ψ q + 1 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 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 SuperscriptBox RowBox ( RowBox - 1 ) SubscriptBox ψ q RowBox UnderoverscriptBox RowBox k = 0 RowBox j - SubscriptBox ψ q RowBox FractionBox RowBox TagBox SubscriptBox RowBox ( RowBox SubscriptBox ψ q - j ) 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 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 TagBox ψ PolyGamma ( RowBox j + 1 ) - RowBox TagBox ψ PolyGamma ( RowBox j + SubscriptBox a 1 ) - RowBox TagBox ψ PolyGamma ( RowBox j + SubscriptBox a 2 ) + RowBox TagBox ψ PolyGamma ( RowBox j - k - SubscriptBox ψ q + 1 ) ) ) /; RowBox RowBox RowBox Re ( SubscriptBox a 3 ) > RowBox - j RowBox RowBox Re ( SubscriptBox a RowBox q + 1 ) > RowBox - j ) RowBox SubscriptBox v j RowBox FractionBox RowBox SuperscriptBox RowBox ( RowBox - 1 ) RowBox SubscriptBox ψ q - 1 TagBox SubscriptBox RowBox ( RowBox SubscriptBox a 1 + SubscriptBox ψ q ) RowBox j - SubscriptBox ψ q Pochhammer TagBox SubscriptBox RowBox ( RowBox SubscriptBox a 2 + SubscriptBox ψ q ) RowBox j - SubscriptBox ψ q Pochhammer RowBox RowBox j ! RowBox RowBox ( RowBox j - SubscriptBox ψ q ) ! RowBox UnderoverscriptBox RowBox k = 0 RowBox j - SubscriptBox ψ q FractionBox RowBox TagBox SubscriptBox RowBox ( RowBox SubscriptBox ψ q - j ) 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 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 - SubscriptBox ψ q TraditionalForm [/itex]

 Rule Form

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

 2001-10-29