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/07.32.06.0009.01

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

 Rule Form

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

 2001-10-29