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variants of this functions
HypergeometricPFQ






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/07.31.06.0008.01









  


  










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}]) ((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







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</mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &lt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> <mo> &#10869; </mo> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </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> 1 </mn> </mrow> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <msub> <mi> a </mi> <mi> j </mi> </msub> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> q </mi> <mo> &gt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> h </mi> <mi> j </mi> </msub> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[SubscriptBox[&quot;a&quot;, &quot;1&quot;], &quot;+&quot;, SubscriptBox[&quot;\[Psi]&quot;, &quot;q&quot;]]], &quot;)&quot;]], &quot;j&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[SubscriptBox[&quot;a&quot;, &quot;2&quot;], &quot;+&quot;, SubscriptBox[&quot;\[Psi]&quot;, &quot;q&quot;]]], &quot;)&quot;]], &quot;j&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> - </mo> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> j </mi> </munderover> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;-&quot;, &quot;j&quot;]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <msubsup> <mi> &#8496; </mi> <mi> k </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> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[SubscriptBox[&quot;a&quot;, &quot;1&quot;], &quot;+&quot;, SubscriptBox[&quot;\[Psi]&quot;, &quot;q&quot;]]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[SubscriptBox[&quot;a&quot;, &quot;2&quot;], &quot;+&quot;, SubscriptBox[&quot;\[Psi]&quot;, &quot;q&quot;]]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> </mrow> </mfrac> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> u </mi> <mi> j </mi> </msub> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> j </mi> <mo> - </mo> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[SubscriptBox[&quot;a&quot;, &quot;1&quot;], &quot;+&quot;, SubscriptBox[&quot;\[Psi]&quot;, &quot;q&quot;]]], &quot;)&quot;]], RowBox[List[&quot;j&quot;, &quot;-&quot;, SubscriptBox[&quot;\[Psi]&quot;, &quot;q&quot;]]]], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> j </mi> <mo> - </mo> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[SubscriptBox[&quot;a&quot;, &quot;2&quot;], &quot;+&quot;, SubscriptBox[&quot;\[Psi]&quot;, &quot;q&quot;]]], &quot;)&quot;]], RowBox[List[&quot;j&quot;, &quot;-&quot;, SubscriptBox[&quot;\[Psi]&quot;, &quot;q&quot;]]]], Pochhammer] </annotation> </semantics> </mrow> <mrow> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> - </mo> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> - </mo> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mrow> <mi> j </mi> <mo> - </mo> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mi> &#8734; </mi> </munderover> <mfrac> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> j </mi> <mo> + </mo> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <msubsup> <mi> &#8496; </mi> <mi> k </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> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[SubscriptBox[&quot;a&quot;, &quot;1&quot;], &quot;+&quot;, SubscriptBox[&quot;\[Psi]&quot;, &quot;q&quot;]]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[SubscriptBox[&quot;a&quot;, &quot;2&quot;], &quot;+&quot;, SubscriptBox[&quot;\[Psi]&quot;, &quot;q&quot;]]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> </mrow> </mfrac> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </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> <mrow> <mi> j </mi> <mo> - </mo> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> </mrow> </munderover> <mrow> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> <mo> - </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[SubscriptBox[&quot;\[Psi]&quot;, &quot;q&quot;], &quot;-&quot;, &quot;j&quot;]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <msubsup> <mi> &#8496; </mi> <mi> k </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> <mtext> </mtext> </mrow> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[SubscriptBox[&quot;a&quot;, &quot;1&quot;], &quot;+&quot;, SubscriptBox[&quot;\[Psi]&quot;, &quot;q&quot;]]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[SubscriptBox[&quot;a&quot;, &quot;2&quot;], &quot;+&quot;, SubscriptBox[&quot;\[Psi]&quot;, &quot;q&quot;]]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> j </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> j </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> j </mi> <mo> - </mo> <mi> k </mi> <mo> - </mo> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> a </mi> <mn> 3 </mn> </msub> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mrow> <mo> - </mo> <mi> j </mi> </mrow> </mrow> <mo> &#8743; </mo> <mo> &#8230; </mo> <mo> &#8743; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> a </mi> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mrow> <mo> - </mo> <mi> j </mi> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> v </mi> <mi> j </mi> </msub> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> j </mi> <mo> - </mo> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[SubscriptBox[&quot;a&quot;, &quot;1&quot;], &quot;+&quot;, SubscriptBox[&quot;\[Psi]&quot;, &quot;q&quot;]]], &quot;)&quot;]], RowBox[List[&quot;j&quot;, &quot;-&quot;, SubscriptBox[&quot;\[Psi]&quot;, &quot;q&quot;]]]], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> j </mi> <mo> - </mo> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[SubscriptBox[&quot;a&quot;, &quot;2&quot;], &quot;+&quot;, SubscriptBox[&quot;\[Psi]&quot;, &quot;q&quot;]]], &quot;)&quot;]], RowBox[List[&quot;j&quot;, &quot;-&quot;, SubscriptBox[&quot;\[Psi]&quot;, &quot;q&quot;]]]], Pochhammer] </annotation> </semantics> </mrow> <mrow> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> - </mo> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> j </mi> <mo> - </mo> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> </mrow> </munderover> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> <mo> - </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[SubscriptBox[&quot;\[Psi]&quot;, &quot;q&quot;], &quot;-&quot;, &quot;j&quot;]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <msubsup> <mi> &#8496; </mi> <mi> k </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> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[SubscriptBox[&quot;a&quot;, &quot;1&quot;], &quot;+&quot;, SubscriptBox[&quot;\[Psi]&quot;, &quot;q&quot;]]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[SubscriptBox[&quot;a&quot;, &quot;2&quot;], &quot;+&quot;, SubscriptBox[&quot;\[Psi]&quot;, &quot;q&quot;]]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> </mrow> </mfrac> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mo> - </mo> <msub> <mi> &#968; </mi> <mi> q </mi> </msub> </mrow> <mo> &#8712; 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Date Added to functions.wolfram.com (modification date)





2001-10-29





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