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; }

 HypergeometricPFQ

 http://functions.wolfram.com/07.31.06.0005.01

 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[Subscript[g, k][0] (1 - z)^k, {k, 0, Infinity}] + (1 - z)^Subscript[\[Psi], q] Sum[Subscript[g, k][Subscript[\[Psi], q]] (1 - z)^k, {k, 0, Infinity}]) /; Abs[1 - z] < 1 && q > 1 && Subscript[g, k][r] == ((-1)^k/k!) Gamma[Subscript[a, 1] + r + k] Gamma[Subscript[a, 2] + r + k] Gamma[Subscript[\[Psi], q] - 2 r - k] Sum[(Pochhammer[Subscript[\[Psi], q] - r - k, j]/ (Gamma[Subscript[\[Psi], q] + Subscript[a, 1] + j] Gamma[Subscript[\[Psi], q] + Subscript[a, 2] + j])) HypergeometricPFQExpansionCoefficient[{Subscript[a, 1], \[Ellipsis], Subscript[a, q + 1]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, j], {j, 0, Infinity}] == (((-1)^k Gamma[Subscript[\[Psi], q] - 2 r - k])/k!) (Product[Gamma[Subscript[a, j] + r + k], {j, 1, q + 1}]/ Product[Gamma[Subscript[b, j] + r + k], {j, 1, q}]) Limit[(1/Gamma[Subscript[\[Psi], q] - r - k]) HypergeometricPFQ[ {Subscript[a, 1] + r + k, Subscript[a, 2] + r + k, \[Ellipsis], Subscript[a, q + 1] + r + n, -m}, {Subscript[b, 1] + r + k, Subscript[b, 2] + r + k, \[Ellipsis], Subscript[b, q] + r + n, 1 - Subscript[\[Psi], q] + r + k - m}, 1], m -> Infinity] && Subscript[g, 0][Subscript[\[Psi], q]] == Gamma[-Subscript[\[Psi], q]] && HypergeometricPFQExpansionCoefficient[{Subscript[a, 1], \[Ellipsis], Subscript[a, q + 1]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, Subscript[k, 1]] == (Pochhammer[Subscript[b, 1] - Subscript[a, 3], Subscript[k, 1]]/Subscript[k, 1]!) Pochhammer[Sum[Subscript[b, j], {j, 2, q}] - Sum[Subscript[a, j], {j, 3, q + 1}], Subscript[k, 1]] Sum[(Pochhammer[-Subscript[k, 1], Subscript[k, 2]]/ (Pochhammer[Sum[Subscript[b, j], {j, 2, q}] - Sum[Subscript[a, j], {j, 3, q + 1}], Subscript[k, 2]] Pochhammer[1 - Subscript[b, 1] + Subscript[a, 3] - Subscript[k, 1], Subscript[k, 2]])) (1/Subscript[k, 2]!) Pochhammer[Sum[Subscript[b, j], {j, 3, q}] - Sum[Subscript[a, j], {j, 4, q + 1}], Subscript[k, 2]] Pochhammer[Subscript[b, 2] - Subscript[a, 4], Subscript[k, 2]] Sum[(Pochhammer[-Subscript[k, 2], Subscript[k, 3]]/ (Pochhammer[Sum[Subscript[b, j], {j, 3, q}] - Sum[Subscript[a, j], {j, 4, q + 1}], Subscript[k, 3]] Pochhammer[ 1 - Subscript[b, 2] + Subscript[a, 4] - Subscript[k, 2], Subscript[k, 3]])) \[Ellipsis] (1/Subscript[k, q - 2]!) Pochhammer[Sum[Subscript[b, j], {j, q - 1, q}] - Sum[Subscript[a, j], {j, q, q + 1}], Subscript[k, q - 2]] Pochhammer[ Subscript[b, q - 2] - Subscript[a, q], Subscript[k, q - 2]] Sum[(Pochhammer[-Subscript[k, q - 2], Subscript[k, q - 1]]/ (Pochhammer[Sum[Subscript[b, j], {j, q - 1, q}] - Sum[Subscript[a, j], {j, q, q + 1}], Subscript[k, q - 1]] Pochhammer[1 - Subscript[b, q - 2] + Subscript[a, q] - Subscript[k, q - 2], Subscript[k, q - 1]])) (1/Subscript[k, q - 1]!) Pochhammer[Subscript[b, q] - Subscript[a, q + 1], Subscript[k, q - 1]] Pochhammer[Subscript[b, q - 1] - Subscript[a, q + 1], Subscript[k, q - 1]], {Subscript[k, q - 1], 0, Subscript[k, q - 2]}], {Subscript[k, 3], 0, Subscript[k, 2]}], {Subscript[k, 2], 0, Subscript[k, 1]}] && Subscript[\[Psi], q] == Sum[Subscript[b, j], {j, 1, q}] - Sum[Subscript[a, j], {j, 1, q + 1}] && !Element[Subscript[\[Psi], q], Integers] && Re[Subscript[a, 3]] > 0 && \[Ellipsis] && Re[Subscript[a, q + 1]] > 0

 Standard Form

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 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 ) ( ( 1 - z ) ψ q k = 0 g k ( ψ q ) ( 1 - z ) k + k = 0 g k ( 0 ) ( 1 - z ) k ) /; "\[LeftBracketingBar]" 1 - z "\[RightBracketingBar]" < 1 q > 1 g k ( r ) ( - 1 ) k Γ ( k + r + a 1 ) Γ ( k + r + a 2 ) Γ ( ψ q - 2 r - k ) k ! j = 0 ( ψ q - r - k ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["\[Psi]", "q"], "-", "r", "-", "k"]], ")"]], "j"], Pochhammer] j ( q ) ( { a 1 , , a q + 1 } , { b 1 , , b q } ) Γ ( j + a 1 + ψ q ) Γ ( j + a 2 + ψ q ) ( - 1 ) k Γ ( ψ q - 2 r - k ) k ! j = 1 q Γ ( k + r + b j ) ( j = 1 q + 1 Γ ( k + r + a j ) ) lim m "\[Rule]" 1 Γ ( ψ q - r - k ) q + 1 F q ( k + r + a 1 , k + r + a 2 , , n + r + a q + 1 , - m TagBox[TagBox[RowBox[List[TagBox[RowBox[List["k", "+", "r", "+", SubscriptBox["a", "1"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["k", "+", "r", "+", SubscriptBox["a", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["n", "+", "r", "+", SubscriptBox["a", RowBox[List["q", "+", "1"]]]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "m"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]] ; k + r + b 1 , k + r + b 2 , , n + r + b q , k - m + r - ψ q + 1 TagBox[TagBox[RowBox[List[TagBox[RowBox[List["k", "+", "r", "+", SubscriptBox["b", "1"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["k", "+", "r", "+", SubscriptBox["b", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["n", "+", "r", "+", SubscriptBox["b", "q"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["k", "-", "m", "+", "r", "-", SubscriptBox["\[Psi]", "q"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]] ; 1 TagBox["1", HypergeometricPFQ, Rule[Editable, True]] ) g 0 ( ψ q ) Γ ( - ψ q ) k 1 ( q ) ( { a 1 , , a q + 1 } , { b 1 , , b q } ) ( b 1 - a 3 ) k 1 k 1 ! ( j = 2 q b j - j = 3 q + 1 a j ) k 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "2"]], "q"], SubscriptBox["b", "j"]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "3"]], RowBox[List["q", "+", "1"]]], SubscriptBox["a", "j"]]]]], ")"]], SubscriptBox["k", "1"]], Pochhammer] k 2 = 0 k 1 ( - k 1 ) k 2 ( j = 2 q b j - j = 3 q + 1 a j ) k 2 ( a 3 - b 1 - k 1 + 1 ) k 2 TagBox[FractionBox[RowBox[List[SubscriptBox[RowBox[List["(", RowBox[List["-", SubscriptBox["k", "1"]]], ")"]], RowBox[List[SubscriptBox["k", "2"], " "]]], " "]], RowBox[List[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "2"]], "q"], SubscriptBox["b", "j"]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "3"]], RowBox[List["q", "+", "1"]]], SubscriptBox["a", "j"]]]]], ")"]], SubscriptBox["k", "2"]], SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "3"], "-", SubscriptBox["b", "1"], "-", SubscriptBox["k", "1"], "+", "1"]], ")"]], SubscriptBox["k", "2"]]]]], Pochhammer] ( b 2 - a 4 ) k 2 k 2 ! ( j = 3 q b j - j = 4 q + 1 a j ) k 2 TagBox[RowBox[List[FractionBox[TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["b", "2"], "-", SubscriptBox["a", "4"]]], ")"]], SubscriptBox["k", "2"]], Pochhammer], RowBox[List[SubscriptBox["k", "2"], "!"]]], SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "3"]], "q"], SubscriptBox["b", "j"]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "4"]], RowBox[List["q", "+", "1"]]], SubscriptBox["a", "j"]]]]], ")"]], SubscriptBox["k", "2"]]]], Pochhammer] k 3 = 0 k 2 ( - k 2 ) k 3 ( j = 3 q b j - j = 4 q + 1 a j ) k 3 ( a 4 - b 2 - k 2 + 1 ) k 3 TagBox[FractionBox[RowBox[List[SubscriptBox[RowBox[List["(", RowBox[List["-", SubscriptBox["k", "2"]]], ")"]], SubscriptBox["k", "3"]], " "]], RowBox[List[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "3"]], "q"], SubscriptBox["b", "j"]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "4"]], RowBox[List["q", "+", "1"]]], SubscriptBox["a", "j"]]]]], ")"]], SubscriptBox["k", "3"]], SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "4"], "-", SubscriptBox["b", "2"], "-", SubscriptBox["k", "2"], "+", "1"]], ")"]], SubscriptBox["k", "3"]]]]], Pochhammer] ( b q - 2 - a q ) k q - 2 TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["b", RowBox[List["q", "-", "2"]]], "-", SubscriptBox["a", "q"]]], ")"]], SubscriptBox["k", RowBox[List["q", "-", "2"]]]], Pochhammer] k q - 2 ! ( j = q - 1 q b j - j = q q + 1 a j ) k q - 2 TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", RowBox[List["q", "-", "1"]]]], "q"], SubscriptBox["b", "j"]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "q"]], RowBox[List["q", "+", "1"]]], SubscriptBox["a", "j"]]]]], ")"]], SubscriptBox["k", RowBox[List["q", "-", "2"]]]], Pochhammer] k q - 1 = 0 k q - 2 ( - k q - 2 ) k q - 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", SubscriptBox["k", RowBox[List["q", "-", "2"]]]]], ")"]], SubscriptBox["k", RowBox[List["q", "-", "1"]]]], Pochhammer] ( j = q - 1 q b j - j = q q + 1 a j ) k q - 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", RowBox[List["q", "-", "1"]]]], "q"], SubscriptBox["b", "j"]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "q"]], RowBox[List["q", "+", "1"]]], SubscriptBox["a", "j"]]]]], ")"]], SubscriptBox["k", RowBox[List["q", "-", "1"]]]], Pochhammer] ( a q - b q - 2 - k q - 2 + 1 ) k q - 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "q"], "-", SubscriptBox["b", RowBox[List["q", "-", "2"]]], "-", SubscriptBox["k", RowBox[List["q", "-", "2"]]], "+", "1"]], ")"]], SubscriptBox["k", RowBox[List["q", "-", "1"]]]], Pochhammer] ( b q - a q + 1 ) k q - 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["b", "q"], "-", SubscriptBox["a", RowBox[List["q", "+", "1"]]]]], ")"]], SubscriptBox["k", RowBox[List["q", "-", "1"]]]], Pochhammer] ( b q - 1 - a q + 1 ) k q - 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["b", RowBox[List["q", "-", "1"]]], "-", SubscriptBox["a", RowBox[List["q", "+", "1"]]]]], ")"]], SubscriptBox["k", RowBox[List["q", "-", "1"]]]], Pochhammer] k q - 2 ! ψ q j = 1 q b j - j = 1 q + 1 a j ψ q TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Re ( a 3 ) > 0 Re ( a q + 1 ) > 0 FormBox RowBox RowBox TagBox TagBox RowBox RowBox SubscriptBox FormBox RowBox q + 1 TraditionalForm SubscriptBox 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 ) InterpretTemplate Function HypergeometricPFQ Slot 1 Slot 2 Slot 3 Rule Editable HypergeometricPFQ RowBox FractionBox RowBox UnderoverscriptBox RowBox k = 1 q RowBox Γ ( SubscriptBox b k ) 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 k = 0 RowBox RowBox SubscriptBox g k ( SubscriptBox ψ q ) SuperscriptBox RowBox ( RowBox 1 - z ) k + RowBox UnderoverscriptBox RowBox k = 0 RowBox RowBox SubscriptBox g k ( 0 ) SuperscriptBox RowBox ( RowBox 1 - z ) k ) /; RowBox RowBox RowBox RowBox 1 - z < 1 RowBox q > 1 RowBox RowBox SubscriptBox g k ( r ) RowBox FractionBox RowBox SuperscriptBox RowBox ( RowBox - 1 ) k RowBox Γ ( RowBox k + r + SubscriptBox a 1 ) RowBox Γ ( RowBox k + r + SubscriptBox a 2 ) RowBox Γ ( RowBox SubscriptBox ψ q - RowBox 2 r - k ) RowBox k ! RowBox UnderoverscriptBox RowBox j = 0 FractionBox RowBox TagBox SubscriptBox RowBox ( RowBox SubscriptBox ψ q - r - k ) j Pochhammer RowBox SubsuperscriptBox j 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 j + SubscriptBox a 1 + SubscriptBox ψ q ) RowBox Γ ( RowBox j + SubscriptBox a 2 + SubscriptBox ψ q ) RowBox FractionBox RowBox SuperscriptBox RowBox ( RowBox - 1 ) k RowBox Γ ( RowBox SubscriptBox ψ q - RowBox 2 r - k ) RowBox RowBox k ! RowBox UnderoverscriptBox RowBox j = 1 q RowBox Γ ( RowBox k + r + SubscriptBox b j ) RowBox ( RowBox UnderoverscriptBox RowBox j = 1 RowBox q + 1 RowBox Γ ( RowBox k + r + SubscriptBox a j ) ) RowBox Limit [ RowBox FractionBox 1 RowBox Γ ( RowBox SubscriptBox ψ q - r - k ) RowBox RowBox SubscriptBox ErrorBox FormBox RowBox q + 1 TraditionalForm SubscriptBox F FormBox q TraditionalForm RowBox ( RowBox TagBox TagBox RowBox TagBox RowBox k + r + SubscriptBox a 1 HypergeometricPFQ Rule Editable , TagBox RowBox k + r + SubscriptBox a 2 HypergeometricPFQ Rule Editable , TagBox HypergeometricPFQ Rule Editable , TagBox RowBox n + r + SubscriptBox a RowBox q + 1 HypergeometricPFQ Rule Editable , TagBox RowBox - m HypergeometricPFQ Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable ; TagBox TagBox RowBox TagBox RowBox k + r + SubscriptBox b 1 HypergeometricPFQ Rule Editable , TagBox RowBox k + r + SubscriptBox b 2 HypergeometricPFQ Rule Editable , TagBox HypergeometricPFQ Rule Editable , TagBox RowBox n + r + SubscriptBox b q HypergeometricPFQ Rule Editable , TagBox RowBox k - m + r - SubscriptBox ψ q + 1 HypergeometricPFQ Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable ; TagBox 1 HypergeometricPFQ Rule Editable ) , RowBox m -> ] RowBox RowBox SubscriptBox g 0 ( SubscriptBox ψ q ) RowBox Γ ( RowBox - SubscriptBox ψ q ) RowBox RowBox SubsuperscriptBox SubscriptBox k 1 RowBox ( q ) ( RowBox RowBox { RowBox SubscriptBox a 1 , , SubscriptBox a RowBox q + 1 } , RowBox { RowBox SubscriptBox b 1 , , SubscriptBox b q } ) RowBox FractionBox SubscriptBox RowBox ( RowBox SubscriptBox b 1 - SubscriptBox a 3 ) SubscriptBox k 1 RowBox SubscriptBox k 1 ! TagBox SubscriptBox RowBox ( RowBox RowBox UnderoverscriptBox RowBox j = 2 q SubscriptBox b j - RowBox UnderoverscriptBox RowBox j = 3 RowBox q + 1 SubscriptBox a j ) SubscriptBox k 1 Pochhammer RowBox UnderoverscriptBox RowBox SubscriptBox k 2 = 0 SubscriptBox k 1 RowBox TagBox FractionBox RowBox SubscriptBox RowBox ( RowBox - SubscriptBox k 1 ) RowBox SubscriptBox k 2 RowBox SubscriptBox RowBox ( RowBox RowBox UnderoverscriptBox RowBox j = 2 q SubscriptBox b j - RowBox UnderoverscriptBox RowBox j = 3 RowBox q + 1 SubscriptBox a j ) SubscriptBox k 2 SubscriptBox RowBox ( RowBox SubscriptBox a 3 - SubscriptBox b 1 - SubscriptBox k 1 + 1 ) SubscriptBox k 2 Pochhammer TagBox RowBox FractionBox TagBox SubscriptBox RowBox ( RowBox SubscriptBox b 2 - SubscriptBox a 4 ) SubscriptBox k 2 Pochhammer RowBox SubscriptBox k 2 ! SubscriptBox RowBox ( RowBox RowBox UnderoverscriptBox RowBox j = 3 q SubscriptBox b j - RowBox UnderoverscriptBox RowBox j = 4 RowBox q + 1 SubscriptBox a j ) SubscriptBox k 2 Pochhammer RowBox UnderoverscriptBox RowBox SubscriptBox k 3 = 0 SubscriptBox k 2 RowBox TagBox FractionBox RowBox SubscriptBox RowBox ( RowBox - SubscriptBox k 2 ) SubscriptBox k 3 RowBox SubscriptBox RowBox ( RowBox RowBox UnderoverscriptBox RowBox j = 3 q SubscriptBox b j - RowBox UnderoverscriptBox RowBox j = 4 RowBox q + 1 SubscriptBox a j ) SubscriptBox k 3 SubscriptBox RowBox ( RowBox SubscriptBox a 4 - SubscriptBox b 2 - SubscriptBox k 2 + 1 ) SubscriptBox k 3 Pochhammer FractionBox TagBox SubscriptBox RowBox ( RowBox SubscriptBox b RowBox q - 2 - SubscriptBox a q ) SubscriptBox k RowBox q - 2 Pochhammer RowBox SubscriptBox k RowBox q - 2 ! TagBox SubscriptBox RowBox ( RowBox RowBox UnderoverscriptBox RowBox j = RowBox q - 1 q SubscriptBox b j - RowBox UnderoverscriptBox RowBox j = q RowBox q + 1 SubscriptBox a j ) SubscriptBox k RowBox q - 2 Pochhammer RowBox UnderoverscriptBox RowBox SubscriptBox k RowBox q - 1 = 0 SubscriptBox k RowBox q - 2 RowBox FractionBox TagBox SubscriptBox RowBox ( RowBox - SubscriptBox k RowBox q - 2 ) SubscriptBox k RowBox q - 1 Pochhammer RowBox TagBox SubscriptBox RowBox ( RowBox RowBox UnderoverscriptBox RowBox j = RowBox q - 1 q SubscriptBox b j - RowBox UnderoverscriptBox RowBox j = q RowBox q + 1 SubscriptBox a j ) SubscriptBox k RowBox q - 1 Pochhammer TagBox SubscriptBox RowBox ( RowBox SubscriptBox a q - SubscriptBox b RowBox q - 2 - SubscriptBox k RowBox q - 2 + 1 ) SubscriptBox k RowBox q - 1 Pochhammer FractionBox RowBox TagBox SubscriptBox RowBox ( RowBox SubscriptBox b q - SubscriptBox a RowBox q + 1 ) SubscriptBox k RowBox q - 1 Pochhammer TagBox SubscriptBox RowBox ( RowBox SubscriptBox b RowBox q - 1 - SubscriptBox a RowBox q + 1 ) SubscriptBox k RowBox q - 1 Pochhammer RowBox SubscriptBox k RowBox q - 2 ! 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 SubscriptBox ψ q TagBox Function RowBox RowBox Re ( SubscriptBox a 3 ) > 0 RowBox RowBox Re ( SubscriptBox a RowBox q + 1 ) > 0 TraditionalForm [/itex]

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

 2001-10-29