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variants of this functions
HypergeometricPFQRegularized






Mathematica Notation

Traditional Notation









Hypergeometric Functions > HypergeometricPFQRegularized[{a1,...,ap},{b1,...,bq},z] > Series representations > Generalized power series > Expansions at z==1 for p==q+1 > The general formulas





http://functions.wolfram.com/07.32.06.0005.01









  


  










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[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] HypergeometricPFQExpansionCoefficient[{Subscript[a, 1], \[Ellipsis], Subscript[a, q + 1]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, j])/(Gamma[Subscript[\[Psi], q] + Subscript[a, 1] + j] Gamma[Subscript[\[Psi], q] + Subscript[a, 2] + 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







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</mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mtext> </mtext> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mrow> <msub> <mi> g </mi> <mi> k </mi> </msub> <mo> ( </mo> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mrow> <msub> <mi> g </mi> <mi> k </mi> </msub> <mo> ( </mo> <mn> 0 </mn> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &lt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> q </mi> <mo> &gt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <msub> <mi> g </mi> <mi> k </mi> </msub> <mo> ( </mo> <mi> r </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> r </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> r </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> r </mi> </mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> <mo> - </mo> <mi> r </mi> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[SubscriptBox[&quot;\[Psi]&quot;, &quot;q&quot;], &quot;-&quot;, &quot;r&quot;, &quot;-&quot;, &quot;k&quot;]], &quot;)&quot;]], &quot;j&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <msubsup> <mi> &#8496; </mi> <mi> j </mi> <mrow> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mrow> <mrow> <mo> { </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> a </mi> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> b </mi> <mi> q </mi> </msub> </mrow> <mo> } </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> j </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> j </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> r </mi> </mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </munderover> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> r </mi> <mo> + </mo> <msub> <mi> b </mi> <mi> j </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> r </mi> <mo> + </mo> <msub> <mi> a </mi> <mi> j </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <munder> <mi> lim </mi> <mrow> <mi> m </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> </munder> <mo> &#8290; </mo> <mtext> &#8201; </mtext> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> <mo> - </mo> <mi> r </mi> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <msub> <mi> F </mi> <mi> q </mi> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <semantics> <mrow> <mrow> <mi> k </mi> <mo> + </mo> <mi> r </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> r </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> r </mi> <mo> + </mo> <msub> <mi> a </mi> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mi> m </mi> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[TagBox[RowBox[List[&quot;k&quot;, &quot;+&quot;, &quot;r&quot;, &quot;+&quot;, SubscriptBox[&quot;a&quot;, &quot;1&quot;]]], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;k&quot;, &quot;+&quot;, &quot;r&quot;, &quot;+&quot;, SubscriptBox[&quot;a&quot;, &quot;2&quot;]]], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;n&quot;, &quot;+&quot;, &quot;r&quot;, &quot;+&quot;, SubscriptBox[&quot;a&quot;, RowBox[List[&quot;q&quot;, &quot;+&quot;, &quot;1&quot;]]]]], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;-&quot;, &quot;m&quot;]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]] </annotation> </semantics> <mo> ; </mo> <semantics> <mrow> <mrow> <mi> k </mi> <mo> + </mo> <mi> r </mi> <mo> + </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> r </mi> <mo> + </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> r </mi> <mo> + </mo> <msub> <mi> b </mi> <mi> q </mi> </msub> </mrow> <mo> , </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> m </mi> <mo> + </mo> <mi> r </mi> <mo> - </mo> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[TagBox[RowBox[List[&quot;k&quot;, &quot;+&quot;, &quot;r&quot;, &quot;+&quot;, SubscriptBox[&quot;b&quot;, &quot;1&quot;]]], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;k&quot;, &quot;+&quot;, &quot;r&quot;, &quot;+&quot;, SubscriptBox[&quot;b&quot;, &quot;2&quot;]]], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;n&quot;, &quot;+&quot;, &quot;r&quot;, &quot;+&quot;, SubscriptBox[&quot;b&quot;, &quot;q&quot;]]], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;k&quot;, &quot;-&quot;, &quot;m&quot;, &quot;+&quot;, &quot;r&quot;, &quot;-&quot;, SubscriptBox[&quot;\[Psi]&quot;, &quot;q&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]] </annotation> </semantics> <mo> ; </mo> <semantics> <mn> 1 </mn> <annotation encoding='Mathematica'> TagBox[&quot;1&quot;, HypergeometricPFQ, Rule[Editable, True]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <msub> <mi> g </mi> <mn> 0 </mn> </msub> <mo> ( </mo> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <msubsup> <mi> &#8496; </mi> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mrow> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mrow> <mrow> <mo> { </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> a </mi> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> b </mi> <mi> q </mi> </msub> </mrow> <mo> } </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <msub> <mrow> <mo> ( </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> a </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> </msub> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 2 </mn> </mrow> <mi> q </mi> </munderover> <msub> <mi> b </mi> <mi> j </mi> </msub> </mrow> <mo> - </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 3 </mn> </mrow> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <msub> <mi> a </mi> <mi> j </mi> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[UnderoverscriptBox[&quot;\[Sum]&quot;, RowBox[List[&quot;j&quot;, &quot;=&quot;, &quot;2&quot;]], &quot;q&quot;], SubscriptBox[&quot;b&quot;, &quot;j&quot;]]], &quot;-&quot;, RowBox[List[UnderoverscriptBox[&quot;\[Sum]&quot;, RowBox[List[&quot;j&quot;, &quot;=&quot;, &quot;3&quot;]], RowBox[List[&quot;q&quot;, &quot;+&quot;, &quot;1&quot;]]], SubscriptBox[&quot;a&quot;, &quot;j&quot;]]]]], &quot;)&quot;]], SubscriptBox[&quot;k&quot;, &quot;1&quot;]], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <mtext> </mtext> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> </munderover> <mrow> <semantics> <mfrac> <mrow> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <msub> <mi> k </mi> <mn> 1 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mrow> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mtext> </mtext> </mrow> </msub> <mtext> </mtext> </mrow> <mrow> <msub> <mrow> <mo> ( </mo> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 2 </mn> </mrow> <mi> q </mi> </munderover> <msub> <mi> b </mi> <mi> j </mi> </msub> </mrow> <mo> - </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 3 </mn> </mrow> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <msub> <mi> a </mi> <mi> j </mi> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> <msub> <mi> k </mi> <mn> 2 </mn> </msub> </msub> <mo> &#8290; </mo> <msub> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 3 </mn> </msub> <mo> - </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <msub> <mi> k </mi> <mn> 2 </mn> </msub> </msub> </mrow> </mfrac> <annotation encoding='Mathematica'> TagBox[FractionBox[RowBox[List[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;-&quot;, SubscriptBox[&quot;k&quot;, &quot;1&quot;]]], &quot;)&quot;]], RowBox[List[SubscriptBox[&quot;k&quot;, &quot;2&quot;], &quot; &quot;]]], &quot; &quot;]], RowBox[List[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[UnderoverscriptBox[&quot;\[Sum]&quot;, RowBox[List[&quot;j&quot;, &quot;=&quot;, &quot;2&quot;]], &quot;q&quot;], SubscriptBox[&quot;b&quot;, &quot;j&quot;]]], &quot;-&quot;, RowBox[List[UnderoverscriptBox[&quot;\[Sum]&quot;, RowBox[List[&quot;j&quot;, &quot;=&quot;, &quot;3&quot;]], RowBox[List[&quot;q&quot;, &quot;+&quot;, &quot;1&quot;]]], SubscriptBox[&quot;a&quot;, &quot;j&quot;]]]]], &quot;)&quot;]], SubscriptBox[&quot;k&quot;, &quot;2&quot;]], SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[SubscriptBox[&quot;a&quot;, &quot;3&quot;], &quot;-&quot;, SubscriptBox[&quot;b&quot;, &quot;1&quot;], &quot;-&quot;, SubscriptBox[&quot;k&quot;, &quot;1&quot;], &quot;+&quot;, &quot;1&quot;]], &quot;)&quot;]], SubscriptBox[&quot;k&quot;, &quot;2&quot;]]]]], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <mrow> <mfrac> <msub> <mrow> <mo> ( </mo> <mrow> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> - </mo> <msub> <mi> a </mi> <mn> 4 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <msub> <mi> k </mi> <mn> 2 </mn> </msub> </msub> <mrow> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <msub> <mrow> <mo> ( </mo> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 3 </mn> </mrow> <mi> q </mi> </munderover> <msub> <mi> b </mi> <mi> j </mi> </msub> </mrow> <mo> - </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 4 </mn> </mrow> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <msub> <mi> a </mi> <mi> j </mi> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> <msub> <mi> k </mi> <mn> 2 </mn> </msub> </msub> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[FractionBox[TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[SubscriptBox[&quot;b&quot;, &quot;2&quot;], &quot;-&quot;, SubscriptBox[&quot;a&quot;, &quot;4&quot;]]], &quot;)&quot;]], SubscriptBox[&quot;k&quot;, &quot;2&quot;]], Pochhammer], RowBox[List[SubscriptBox[&quot;k&quot;, &quot;2&quot;], &quot;!&quot;]]], SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[UnderoverscriptBox[&quot;\[Sum]&quot;, RowBox[List[&quot;j&quot;, &quot;=&quot;, &quot;3&quot;]], &quot;q&quot;], SubscriptBox[&quot;b&quot;, &quot;j&quot;]]], &quot;-&quot;, RowBox[List[UnderoverscriptBox[&quot;\[Sum]&quot;, RowBox[List[&quot;j&quot;, &quot;=&quot;, &quot;4&quot;]], RowBox[List[&quot;q&quot;, &quot;+&quot;, &quot;1&quot;]]], SubscriptBox[&quot;a&quot;, &quot;j&quot;]]]]], &quot;)&quot;]], SubscriptBox[&quot;k&quot;, &quot;2&quot;]]]], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mn> 3 </mn> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <msub> <mi> k </mi> <mn> 2 </mn> </msub> </munderover> <mrow> <semantics> <mfrac> <mrow> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <msub> <mi> k </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <msub> <mi> k </mi> <mn> 3 </mn> </msub> </msub> <mtext> </mtext> </mrow> <mrow> <msub> <mrow> <mo> ( </mo> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 3 </mn> </mrow> <mi> q </mi> </munderover> <msub> <mi> b </mi> <mi> j </mi> </msub> </mrow> <mo> - </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 4 </mn> </mrow> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <msub> <mi> a </mi> <mi> j </mi> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> <msub> <mi> k </mi> <mn> 3 </mn> </msub> </msub> <mo> &#8290; </mo> <msub> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 4 </mn> </msub> <mo> - </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> - </mo> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <msub> <mi> k </mi> <mn> 3 </mn> </msub> </msub> </mrow> </mfrac> <annotation encoding='Mathematica'> TagBox[FractionBox[RowBox[List[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;-&quot;, SubscriptBox[&quot;k&quot;, &quot;2&quot;]]], &quot;)&quot;]], SubscriptBox[&quot;k&quot;, &quot;3&quot;]], &quot; &quot;]], RowBox[List[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[UnderoverscriptBox[&quot;\[Sum]&quot;, RowBox[List[&quot;j&quot;, &quot;=&quot;, &quot;3&quot;]], &quot;q&quot;], SubscriptBox[&quot;b&quot;, &quot;j&quot;]]], &quot;-&quot;, RowBox[List[UnderoverscriptBox[&quot;\[Sum]&quot;, RowBox[List[&quot;j&quot;, &quot;=&quot;, &quot;4&quot;]], RowBox[List[&quot;q&quot;, &quot;+&quot;, &quot;1&quot;]]], SubscriptBox[&quot;a&quot;, &quot;j&quot;]]]]], &quot;)&quot;]], SubscriptBox[&quot;k&quot;, &quot;3&quot;]], SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[SubscriptBox[&quot;a&quot;, &quot;4&quot;], &quot;-&quot;, SubscriptBox[&quot;b&quot;, &quot;2&quot;], &quot;-&quot;, SubscriptBox[&quot;k&quot;, &quot;2&quot;], &quot;+&quot;, &quot;1&quot;]], &quot;)&quot;]], SubscriptBox[&quot;k&quot;, &quot;3&quot;]]]]], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <mtext> </mtext> <mo> &#8230; </mo> <mo> &#8290; </mo> <mfrac> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <msub> <mi> b </mi> <mrow> <mi> q </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msub> <mo> - </mo> <msub> <mi> a </mi> <mi> q </mi> </msub> </mrow> <mo> ) </mo> </mrow> <msub> <mi> k </mi> <mrow> <mi> q </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msub> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[SubscriptBox[&quot;b&quot;, RowBox[List[&quot;q&quot;, &quot;-&quot;, &quot;2&quot;]]], &quot;-&quot;, SubscriptBox[&quot;a&quot;, &quot;q&quot;]]], &quot;)&quot;]], SubscriptBox[&quot;k&quot;, RowBox[List[&quot;q&quot;, &quot;-&quot;, &quot;2&quot;]]]], Pochhammer] </annotation> </semantics> <mrow> <msub> <mi> k </mi> <mrow> <mi> q </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msub> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mi> q </mi> </munderover> <msub> <mi> b </mi> <mi> j </mi> </msub> </mrow> <mo> - </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mi> q </mi> </mrow> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <msub> <mi> a </mi> <mi> j </mi> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> <msub> <mi> k </mi> <mrow> <mi> q </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msub> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[UnderoverscriptBox[&quot;\[Sum]&quot;, RowBox[List[&quot;j&quot;, &quot;=&quot;, RowBox[List[&quot;q&quot;, &quot;-&quot;, &quot;1&quot;]]]], &quot;q&quot;], SubscriptBox[&quot;b&quot;, &quot;j&quot;]]], &quot;-&quot;, RowBox[List[UnderoverscriptBox[&quot;\[Sum]&quot;, RowBox[List[&quot;j&quot;, &quot;=&quot;, &quot;q&quot;]], RowBox[List[&quot;q&quot;, &quot;+&quot;, &quot;1&quot;]]], SubscriptBox[&quot;a&quot;, &quot;j&quot;]]]]], &quot;)&quot;]], SubscriptBox[&quot;k&quot;, RowBox[List[&quot;q&quot;, &quot;-&quot;, &quot;2&quot;]]]], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <msub> <mi> k </mi> <mrow> <mi> q </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msub> </munderover> <mrow> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <msub> <mi> k </mi> <mrow> <mi> q </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <msub> <mi> k </mi> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;-&quot;, SubscriptBox[&quot;k&quot;, RowBox[List[&quot;q&quot;, &quot;-&quot;, &quot;2&quot;]]]]], &quot;)&quot;]], SubscriptBox[&quot;k&quot;, RowBox[List[&quot;q&quot;, &quot;-&quot;, &quot;1&quot;]]]], Pochhammer] </annotation> </semantics> <mtext> </mtext> </mrow> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mi> q </mi> </munderover> <msub> <mi> b </mi> <mi> j </mi> </msub> </mrow> <mo> - </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mi> q </mi> </mrow> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <msub> <mi> a </mi> <mi> j </mi> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> <msub> <mi> k </mi> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[UnderoverscriptBox[&quot;\[Sum]&quot;, RowBox[List[&quot;j&quot;, &quot;=&quot;, RowBox[List[&quot;q&quot;, &quot;-&quot;, &quot;1&quot;]]]], &quot;q&quot;], SubscriptBox[&quot;b&quot;, &quot;j&quot;]]], &quot;-&quot;, RowBox[List[UnderoverscriptBox[&quot;\[Sum]&quot;, RowBox[List[&quot;j&quot;, &quot;=&quot;, &quot;q&quot;]], RowBox[List[&quot;q&quot;, &quot;+&quot;, &quot;1&quot;]]], SubscriptBox[&quot;a&quot;, &quot;j&quot;]]]]], &quot;)&quot;]], SubscriptBox[&quot;k&quot;, RowBox[List[&quot;q&quot;, &quot;-&quot;, &quot;1&quot;]]]], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mi> q </mi> </msub> <mo> - </mo> <msub> <mi> b </mi> <mrow> <mi> q </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msub> <mo> - </mo> <msub> <mi> k </mi> <mrow> <mi> q </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <msub> <mi> k </mi> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[SubscriptBox[&quot;a&quot;, &quot;q&quot;], &quot;-&quot;, SubscriptBox[&quot;b&quot;, RowBox[List[&quot;q&quot;, &quot;-&quot;, &quot;2&quot;]]], &quot;-&quot;, SubscriptBox[&quot;k&quot;, RowBox[List[&quot;q&quot;, &quot;-&quot;, &quot;2&quot;]]], &quot;+&quot;, &quot;1&quot;]], &quot;)&quot;]], SubscriptBox[&quot;k&quot;, RowBox[List[&quot;q&quot;, &quot;-&quot;, &quot;1&quot;]]]], Pochhammer] </annotation> </semantics> </mrow> </mfrac> <mo> &#8290; </mo> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> - </mo> <msub> <mi> a </mi> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <msub> <mi> k </mi> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[SubscriptBox[&quot;b&quot;, &quot;q&quot;], &quot;-&quot;, SubscriptBox[&quot;a&quot;, RowBox[List[&quot;q&quot;, &quot;+&quot;, &quot;1&quot;]]]]], &quot;)&quot;]], SubscriptBox[&quot;k&quot;, RowBox[List[&quot;q&quot;, &quot;-&quot;, &quot;1&quot;]]]], Pochhammer] </annotation> </semantics> <mo> &#8290; 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</ms> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> q </ms> </apply> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <list> <apply> <ci> SlotSequence </ci> <cn type='integer'> 1 </cn> </apply> </list> </apply> </apply> </apply> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> ; </ms> <apply> <ci> TagBox </ci> <ms> z </ms> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8719; </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> = </ms> <ms> 1 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> k </ms> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> z </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SubscriptBox </ci> <ms> &#968; </ms> <ms> q </ms> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> = </ms> <ms> 0 </ms> </list> </apply> <ms> &#8734; </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> g </ms> <ms> k </ms> </apply> <ms> ( </ms> <apply> <ci> SubscriptBox </ci> <ms> &#968; </ms> <ms> q </ms> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> z </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> k </ms> </apply> </list> </apply> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> = </ms> <ms> 0 </ms> </list> </apply> <ms> &#8734; </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> g </ms> <ms> k </ms> </apply> <ms> ( </ms> <ms> 0 </ms> <ms> ) </ms> </list> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> z </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> k </ms> </apply> </list> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> &#62979; </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> z </ms> </list> </apply> <ms> &#62980; </ms> </list> </apply> <ms> &lt; </ms> <ms> 1 </ms> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> &gt; </ms> <ms> 1 </ms> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> g </ms> <ms> k </ms> </apply> <ms> ( </ms> <ms> r </ms> <ms> ) </ms> </list> </apply> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> k </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> + </ms> <ms> r </ms> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> 1 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> + </ms> <ms> r </ms> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> &#968; </ms> <ms> q </ms> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> r </ms> </list> </apply> <ms> - </ms> <ms> k </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> ! </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> = </ms> <ms> 0 </ms> </list> </apply> <ms> &#8734; </ms> </apply> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> &#968; </ms> <ms> q </ms> </apply> <ms> - </ms> <ms> r </ms> <ms> - </ms> <ms> k </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> j </ms> </apply> <ci> Pochhammer </ci> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubsuperscriptBox </ci> <ms> &#8496; </ms> <ms> j </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <ms> q </ms> <ms> ) </ms> </list> </apply> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> { </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> 1 </ms> </apply> <ms> , </ms> <ms> &#8230; </ms> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> </list> </apply> <ms> } </ms> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <ms> { </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> 1 </ms> </apply> <ms> , </ms> <ms> &#8230; </ms> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> q </ms> </apply> </list> </apply> <ms> } </ms> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> 1 </ms> </apply> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <ms> &#968; </ms> <ms> q </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> 2 </ms> </apply> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <ms> &#968; </ms> <ms> q </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </apply> </list> </apply> </list> </apply> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> k </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> &#968; </ms> <ms> q </ms> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> r </ms> </list> </apply> <ms> - </ms> <ms> k </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> ! </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8719; </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> = </ms> <ms> 1 </ms> </list> </apply> <ms> q </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> + </ms> <ms> r </ms> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> j </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8719; </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> = </ms> <ms> 1 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> + </ms> <ms> r </ms> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> j </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderscriptBox </ci> <ms> lim </ms> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> &#62754; </ms> <ms> &#8734; </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> &#968; </ms> <ms> q </ms> </apply> <ms> - </ms> <ms> r </ms> <ms> - </ms> <ms> k </ms> </list> </apply> <ms> ) </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> &#62387; </ms> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ci> TraditionalForm </ci> </apply> </apply> <apply> <ci> SubscriptBox </ci> <ms> F </ms> <apply> <ci> FormBox </ci> <ms> q </ms> <ci> TraditionalForm </ci> </apply> </apply> </list> </apply> <ms> &#8289; </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> + </ms> <ms> r </ms> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> 1 </ms> </apply> </list> </apply> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> + </ms> <ms> r </ms> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> 2 </ms> </apply> </list> </apply> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <ms> &#8230; </ms> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> + </ms> <ms> r </ms> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> </list> </apply> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> m </ms> </list> </apply> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <list> <apply> <ci> SlotSequence </ci> <cn type='integer'> 1 </cn> </apply> </list> </apply> </apply> </apply> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> ; </ms> <apply> <ci> TagBox </ci> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> + </ms> <ms> r </ms> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> 1 </ms> </apply> </list> </apply> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> + </ms> <ms> r </ms> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> 2 </ms> </apply> </list> </apply> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <ms> &#8230; </ms> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> + </ms> <ms> r </ms> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> q </ms> </apply> </list> </apply> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> - </ms> <ms> m </ms> <ms> + </ms> <ms> r </ms> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> &#968; </ms> <ms> q </ms> </apply> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <list> <apply> <ci> SlotSequence </ci> <cn type='integer'> 1 </cn> </apply> </list> </apply> </apply> </apply> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> ; </ms> <apply> <ci> TagBox </ci> <ms> 1 </ms> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> g </ms> <ms> 0 </ms> </apply> <ms> ( </ms> <apply> <ci> SubscriptBox </ci> <ms> &#968; </ms> <ms> q </ms> </apply> <ms> ) </ms> </list> </apply> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> &#968; </ms> <ms> q </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubsuperscriptBox </ci> <ms> &#8496; </ms> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 1 </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <ms> q </ms> <ms> ) </ms> </list> </apply> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> { </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> 1 </ms> </apply> <ms> , </ms> <ms> &#8230; </ms> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> </list> </apply> <ms> } </ms> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <ms> { </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> 1 </ms> </apply> <ms> , </ms> <ms> &#8230; </ms> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> q </ms> </apply> </list> </apply> <ms> } </ms> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> SubscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> 1 </ms> </apply> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> 3 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 1 </ms> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 1 </ms> </apply> <ms> ! </ms> </list> </apply> </apply> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> = </ms> <ms> 2 </ms> </list> </apply> <ms> q </ms> </apply> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> j </ms> </apply> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> = </ms> <ms> 3 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> j </ms> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 1 </ms> </apply> </apply> <ci> Pochhammer </ci> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 2 </ms> </apply> <ms> = </ms> <ms> 0 </ms> </list> </apply> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 1 </ms> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 1 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 2 </ms> </apply> <ms> </ms> </list> </apply> </apply> <ms> </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> = </ms> <ms> 2 </ms> </list> </apply> <ms> q </ms> </apply> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> j </ms> </apply> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> = </ms> <ms> 3 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> j </ms> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 2 </ms> </apply> </apply> <apply> <ci> SubscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> 3 </ms> </apply> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> 1 </ms> </apply> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 1 </ms> </apply> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 2 </ms> </apply> </apply> </list> </apply> </apply> <ci> Pochhammer </ci> </apply> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> 2 </ms> </apply> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> 4 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 2 </ms> </apply> </apply> <ci> Pochhammer </ci> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 2 </ms> </apply> <ms> ! </ms> </list> </apply> </apply> <apply> <ci> SubscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> = </ms> <ms> 3 </ms> </list> </apply> <ms> q </ms> </apply> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> j </ms> </apply> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> = </ms> <ms> 4 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> j </ms> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 2 </ms> </apply> </apply> </list> </apply> <ci> Pochhammer </ci> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 3 </ms> </apply> <ms> = </ms> <ms> 0 </ms> </list> </apply> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 2 </ms> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 3 </ms> </apply> </apply> <ms> </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> = </ms> <ms> 3 </ms> </list> </apply> <ms> q </ms> </apply> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> j </ms> </apply> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> = </ms> <ms> 4 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> j </ms> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 3 </ms> </apply> </apply> <apply> <ci> SubscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> 4 </ms> </apply> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> 2 </ms> </apply> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 2 </ms> </apply> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 3 </ms> </apply> </apply> </list> </apply> </apply> <ci> Pochhammer </ci> </apply> <ms> &#8230; </ms> <apply> <ci> FractionBox </ci> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> - </ms> <ms> 2 </ms> </list> </apply> </apply> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> q </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> - </ms> <ms> 2 </ms> </list> </apply> </apply> </apply> <ci> Pochhammer </ci> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> - </ms> <ms> 2 </ms> </list> </apply> </apply> <ms> ! </ms> </list> </apply> </apply> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> = </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> </list> </apply> <ms> q </ms> </apply> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> j </ms> </apply> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> = </ms> <ms> q </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> j </ms> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> - </ms> <ms> 2 </ms> </list> </apply> </apply> </apply> <ci> Pochhammer </ci> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> </apply> <ms> = </ms> <ms> 0 </ms> </list> </apply> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> - </ms> <ms> 2 </ms> </list> </apply> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> - </ms> <ms> 2 </ms> </list> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> </apply> </apply> <ci> Pochhammer </ci> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> = </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> </list> </apply> <ms> q </ms> </apply> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> j </ms> </apply> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> = </ms> <ms> q </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> j </ms> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> </apply> </apply> <ci> Pochhammer </ci> </apply> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> q </ms> </apply> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> - </ms> <ms> 2 </ms> </list> </apply> </apply> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> - </ms> <ms> 2 </ms> </list> </apply> </apply> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> </apply> </apply> <ci> Pochhammer </ci> </apply> </list> </apply> </apply> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> q </ms> </apply> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> </apply> </apply> <ci> Pochhammer </ci> </apply> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> </apply> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> </apply> </apply> <ci> Pochhammer </ci> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> - </ms> <ms> 2 </ms> </list> </apply> </apply> <ms> ! </ms> </list> </apply> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> &#968; </ms> <ms> q </ms> </apply> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> = </ms> <ms> 1 </ms> </list> </apply> <ms> q </ms> </apply> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> j </ms> </apply> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> = </ms> <ms> 1 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> j </ms> </apply> </list> </apply> </list> </apply> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> &#968; </ms> <ms> q </ms> </apply> <ms> &#8713; </ms> <apply> <ci> TagBox </ci> <ms> &#8484; </ms> <apply> <ci> Function </ci> <integers /> </apply> </apply> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> Re </ms> <ms> ( </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> 3 </ms> </apply> <ms> ) </ms> </list> </apply> <ms> &gt; </ms> <ms> 0 </ms> </list> </apply> <ms> &#8743; </ms> <ms> &#8230; </ms> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> Re </ms> <ms> ( </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <ms> ) </ms> </list> </apply> <ms> &gt; </ms> <ms> 0 </ms> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>










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





2001-10-29