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variants of this functions
PolyGamma






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > PolyGamma[nu,z] > Specific values > Values at fixed points





http://functions.wolfram.com/06.15.03.0052.01









  


  










Input Form





PolyGamma[-2 n, p/q] == (1/(-1 + 2 n)!) (((-(p/q)^(-1 + 2 n)) (EulerGamma + (EulerGamma p)/(2 n q) - Pi I - Log[p/q] + PolyGamma[2 n] + Sum[(Binomial[-1 + 2 n, k] (Sum[(p/q)^j Binomial[k, j] PolyGamma[1 - j + k] (2 Zeta[j - k, 1 - p/q] - Zeta[j - k, 1 - p/q]), {j, 0, k}] - PolyGamma[1 + k] Zeta[-k] - Derivative[1][Zeta][-k]))/(-(p/q))^k, {k, 0, -1 + 2 n}]) - (BernoulliB[2 n] (-Log[2 Pi] + PolyGamma[2 n]))/ (q^(2 n) (2 n)) + (BernoulliB[2 n, (-p + q)/q] (-Log[2 Pi q] + PolyGamma[2 n]))/(2 n) + ((-1)^(1 + n) Pi^(1 - 2 n) Sum[PolyGamma[-1 + 2 n, j/q] Sin[(2 j Pi (-p + q))/q], {j, 1, -1 + q}])/(2^(2 n) q^(2 n)) + ((-1)^(1 + n) 2^(1 - 2 n) (-1 + 2 n)! Sum[Cos[(2 j Pi (-p + q))/q] Derivative[1, 0][Zeta][2 n, j/q], {j, 1, -1 + q}])/ (Pi^(2 n) q^(2 n)) + Derivative[1][Zeta][1 - 2 n]/q^(2 n)) - (I (-1 + 2 n) Pi (p/q)^(2 n))/(2 n) - (p/q)^(-1 + 2 n) (-2 EulerGamma + Log[p/(Pi q)] + Pi I + Log[Sin[(p Pi)/q]] - 2 PolyGamma[2 n] + Sum[((-1)^j Binomial[-1 + 2 n, j])/(-1 - j + 2 n), {j, 0, -2 + 2 n}] + Sum[(-1)^j Binomial[-1 + 2 n, j] Sum[(Binomial[-1 - j + 2 n, k] k! PolyLog[1 + k, E^((2 I p Pi)/q)])/ (-((2 Pi I p)/q))^k, {k, 0, -1 - j + 2 n}], {j, 0, -2 + 2 n}]) - (2 I Pi)^(1 - 2 n) Sum[((2 Pi I p)/q)^j Binomial[-1 + 2 n, j] (-1 - j + 2 n)! Zeta[-j + 2 n], {j, 0, -2 + 2 n}]) /; Element[n, Integers] && n > 0 && Element[p, Integers] && 0 < p < q && Element[q, Integers] && q > 1










Standard Form





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MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> p </mi> <mi> q </mi> </mfrac> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; 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</mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> p </mi> </mrow> <mi> q </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mi> j </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mi> k </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[RowBox[List[RowBox[List[&quot;2&quot;, &quot;n&quot;]], &quot;-&quot;, &quot;j&quot;, &quot;-&quot;, &quot;1&quot;]], Identity, Rule[Editable, True]]], List[TagBox[&quot;k&quot;, Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> p </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mi> q </mi> </mfrac> </msup> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </mrow> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </munderover> <mtext> </mtext> <mrow> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> p </mi> </mrow> <mi> q </mi> </mfrac> <mo> ) </mo> </mrow> <mi> j </mi> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mi> j </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[RowBox[List[RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;n&quot;]], &quot;-&quot;, &quot;1&quot;]], Identity, Rule[Editable, True]]], List[TagBox[&quot;j&quot;, Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mi> j </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[RowBox[List[RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;n&quot;]], &quot;-&quot;, &quot;j&quot;]], Zeta, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> n </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> q </mi> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> n </mi> </mrow> </msup> <mo> &#8290; </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> <mrow> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> j </mi> <mi> q </mi> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> j </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> q </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> n </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> q </mi> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> n </mi> </mrow> </msup> <mo> &#8290; </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> <mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> j </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> q </mi> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> &#950; </mi> <semantics> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;1&quot;, &quot;,&quot;, &quot;0&quot;]], &quot;)&quot;]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> , </mo> <mfrac> <mi> j </mi> <mi> q </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <msup> <mi> &#950; </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> q </mi> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> n </mi> </mrow> </msup> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mtext> </mtext> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> q </mi> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> n </mi> </mrow> </msup> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </mfrac> <mo> &#8290; </mo> <msub> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox[&quot;B&quot;, BernoulliB] </annotation> </semantics> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </msub> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox[&quot;B&quot;, BernoulliB] </annotation> </semantics> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </msub> <mo> ( </mo> <mfrac> <mrow> <mi> q </mi> <mo> - </mo> <mi> p </mi> </mrow> <mi> q </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> - </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> p </mi> <mi> q </mi> </mfrac> <mo> ) </mo> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[List[], EulerGamma]] </annotation> </semantics> <mo> + </mo> <mfrac> <mrow> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[List[], EulerGamma]] </annotation> </semantics> <mo> &#8290; </mo> <mi> p </mi> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> <mo> &#8290; </mo> <mi> q </mi> </mrow> </mfrac> <mo> - </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> p </mi> <mi> q </mi> </mfrac> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mi> p </mi> <mi> q </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mi> k </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[RowBox[List[RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;n&quot;]], &quot;-&quot;, &quot;1&quot;]], Identity, Rule[Editable, True]]], List[TagBox[&quot;k&quot;, Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> k </mi> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> p </mi> <mi> q </mi> </mfrac> <mo> ) </mo> </mrow> <mi> j </mi> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> k </mi> </mtd> </mtr> <mtr> <mtd> <mi> j </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[&quot;k&quot;, Identity, Rule[Editable, True]]], List[TagBox[&quot;j&quot;, Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> j </mi> <mo> - </mo> <mi> k </mi> </mrow> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mi> p </mi> <mi> q </mi> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, RowBox[List[TagBox[RowBox[List[&quot;j&quot;, &quot;-&quot;, &quot;k&quot;]], Zeta, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, FractionBox[&quot;p&quot;, &quot;q&quot;]]], Zeta, Rule[Editable, True]]]], &quot;)&quot;]], InterpretTemplate[Function[List[ZetaDump`e1$, ZetaDump`e2$], Zeta[ZetaDump`e1$, ZetaDump`e2$]]]] </annotation> </semantics> </mrow> <mo> - </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> j </mi> <mo> - </mo> <mi> k </mi> </mrow> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mi> p </mi> <mi> q </mi> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, RowBox[List[TagBox[RowBox[List[&quot;j&quot;, &quot;-&quot;, &quot;k&quot;]], Zeta, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, FractionBox[&quot;p&quot;, &quot;q&quot;]]], Zeta, Rule[Editable, True]]]], &quot;)&quot;]], InterpretTemplate[Function[List[ZetaDump`e1$, ZetaDump`e2$], Zeta[ZetaDump`e1$, ZetaDump`e2$]]]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[RowBox[List[&quot;-&quot;, &quot;k&quot;]], Zeta, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <msup> <mi> &#950; </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <msup> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[List[], Integers]] </annotation> </semantics> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mi> p </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mn> 0 </mn> <mo> &lt; </mo> <mi> p </mi> <mo> &lt; </mo> <mi> q </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> q </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> q </mi> <mo> &gt; </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <apply> <ci> TagBox </ci> <ms> &#968; </ms> <ci> PolyGamma </ci> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 2 </ms> </list> </apply> <ms> n </ms> </list> </apply> <ms> ) </ms> </list> </apply> </apply> <ms> ( </ms> <apply> <ci> FractionBox </ci> <ms> p </ms> <ms> q </ms> </apply> <ms> ) </ms> </list> </apply> <ms> &#63449; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> n </ms> </list> </apply> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> ! </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> &#8520; </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> n </ms> </list> </apply> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> &#960; </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> n </ms> </list> </apply> </apply> </list> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> FractionBox </ci> <ms> p </ms> <ms> q </ms> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> n </ms> </list> </apply> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <apply> <ci> FractionBox </ci> <ms> p </ms> <ms> q </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> n </ms> </list> </apply> <ms> - </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> log </ms> <ms> ( </ms> <apply> <ci> FractionBox </ci> <ms> p </ms> <apply> <ci> RowBox </ci> <list> <ms> &#960; </ms> <ms> q </ms> </list> </apply> </apply> <ms> ) </ms> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <ms> &#960; </ms> <ms> &#8520; </ms> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <ms> log </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> sin </ms> <ms> ( </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> p </ms> <ms> &#960; </ms> </list> </apply> <ms> q </ms> </apply> <ms> ) </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> &#968; </ms> <ci> PolyGamma </ci> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> n </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <apply> <ci> TagBox </ci> <ms> &#8509; </ms> <apply> <ci> Function </ci> <list /> <eulergamma /> </apply> </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> 0 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> n </ms> </list> </apply> <ms> - </ms> <ms> 2 </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <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> j </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> n </ms> </list> </apply> <ms> - </ms> <ms> j </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> GridBox </ci> <list> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> </ms> <ms> n </ms> </list> </apply> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ident /> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> <list> <apply> <ci> TagBox </ci> <ms> j </ms> <ident /> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <apply> <ci> Binomial </ci> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Slot </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> </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> j </ms> <ms> = </ms> <ms> 0 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> n </ms> </list> </apply> <ms> - </ms> <ms> 2 </ms> </list> </apply> </apply> <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> j </ms> </apply> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> GridBox </ci> <list> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> </ms> <ms> n </ms> </list> </apply> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ident /> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> <list> <apply> <ci> TagBox </ci> <ms> j </ms> <ident /> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <apply> <ci> Binomial </ci> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Slot </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </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> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> n </ms> </list> </apply> <ms> - </ms> <ms> j </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> &#960; </ms> <ms> &#8520; </ms> <ms> p </ms> </list> </apply> <ms> q </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> k </ms> </list> </apply> </apply> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> GridBox </ci> <list> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> n </ms> </list> </apply> <ms> - </ms> <ms> j </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ident /> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> <list> <apply> <ci> TagBox </ci> <ms> k </ms> <ident /> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <apply> <ci> Binomial </ci> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Slot </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> ! </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <apply> <ci> InterpretationBox </ci> <ms> Li </ms> <ci> PolyLog </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> <apply> <ci> Rule </ci> <ci> Selectable </ci> <false /> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <ms> ( </ms> <apply> <ci> SuperscriptBox </ci> <ms> &#8519; </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> &#8520; </ms> <ms> p </ms> <ms> &#960; </ms> </list> </apply> <ms> q </ms> </apply> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> &#8520; </ms> <ms> &#960; </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> n </ms> </list> </apply> </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> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> n </ms> </list> </apply> <ms> - </ms> <ms> 2 </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> &#960; </ms> <ms> &#8520; </ms> <ms> p </ms> </list> </apply> <ms> q </ms> </apply> <ms> ) </ms> </list> </apply> <ms> j </ms> </apply> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> GridBox </ci> <list> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> </ms> <ms> n </ms> </list> </apply> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ident /> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> <list> <apply> <ci> TagBox </ci> <ms> j </ms> <ident /> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <apply> <ci> Binomial </ci> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Slot </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> n </ms> </list> </apply> <ms> - </ms> <ms> j </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> ! </ms> </list> </apply> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> &#950; </ms> <ms> ( </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> </ms> <ms> n </ms> </list> </apply> <ms> - </ms> <ms> j </ms> </list> </apply> <ci> Zeta </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <ci> BoxForm`e$ </ci> <apply> <ci> Zeta </ci> <ci> BoxForm`e$ </ci> </apply> </apply> </apply> </apply> </list> </apply> </list> </apply> </list> </apply> <ms> + </ms> <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> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> SuperscriptBox </ci> <ms> 2 </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 2 </ms> </list> </apply> <ms> n </ms> </list> </apply> </apply> <apply> <ci> SuperscriptBox </ci> <ms> &#960; </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> n </ms> </list> </apply> </list> </apply> </apply> <apply> <ci> SuperscriptBox </ci> <ms> q </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 2 </ms> </list> </apply> <ms> n </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> 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> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <apply> <ci> TagBox </ci> <ms> &#968; </ms> <ci> PolyGamma </ci> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> n </ms> </list> </apply> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> </apply> <ms> ( </ms> <apply> <ci> FractionBox </ci> <ms> j </ms> <ms> q </ms> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> sin </ms> <ms> ( </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> j </ms> <ms> &#960; </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> - </ms> <ms> p </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> q </ms> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ms> + </ms> <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> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> SuperscriptBox </ci> <ms> 2 </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> n </ms> </list> </apply> </list> </apply> </apply> <apply> <ci> SuperscriptBox </ci> <ms> &#960; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 2 </ms> </list> </apply> <ms> n </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> n </ms> </list> </apply> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> ! </ms> </list> </apply> <apply> <ci> SuperscriptBox </ci> <ms> q </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 2 </ms> </list> </apply> <ms> n </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> 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> <apply> <ci> RowBox </ci> <list> <ms> cos </ms> <ms> ( </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> j </ms> <ms> &#960; </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> - </ms> <ms> p </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> q </ms> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> &#950; </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> , </ms> <ms> 0 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ci> Derivative </ci> </apply> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> n </ms> </list> </apply> <ms> , </ms> <apply> <ci> FractionBox </ci> <ms> j </ms> <ms> q </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> &#950; </ms> <ms> &#8242; </ms> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> n </ms> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SuperscriptBox </ci> <ms> q </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 2 </ms> </list> </apply> <ms> n </ms> </list> </apply> </apply> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> &#968; </ms> <ci> PolyGamma </ci> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> n </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <ms> log </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> &#960; </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SuperscriptBox </ci> <ms> q </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 2 </ms> </list> </apply> <ms> n </ms> </list> </apply> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> n </ms> </list> </apply> </apply> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> B </ms> <ci> BernoulliB </ci> </apply> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> n </ms> </list> </apply> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> ErrorBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> &#968; </ms> <ci> PolyGamma </ci> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> n </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <ms> log </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> &#960; </ms> <ms> q </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> n </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> B </ms> <ci> BernoulliB </ci> </apply> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> n </ms> </list> </apply> </apply> <ms> ( </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> - </ms> <ms> p </ms> </list> </apply> <ms> q </ms> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> FractionBox </ci> <ms> p </ms> <ms> q </ms> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> n </ms> </list> </apply> <ms> - </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> &#8509; </ms> <apply> <ci> Function </ci> <list /> <eulergamma /> </apply> </apply> <ms> + </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> &#8509; </ms> <apply> <ci> Function </ci> <list /> <eulergamma /> </apply> </apply> <ms> p </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> n </ms> <ms> q </ms> </list> </apply> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <ms> &#960; </ms> <ms> &#8520; </ms> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <ms> log </ms> <ms> ( </ms> <apply> <ci> FractionBox </ci> <ms> p </ms> <ms> q </ms> </apply> <ms> ) </ms> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> &#968; </ms> <ci> PolyGamma </ci> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> n </ms> </list> </apply> <ms> ) </ms> </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> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> n </ms> </list> </apply> <ms> - </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <apply> <ci> FractionBox </ci> <ms> p </ms> <ms> q </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> k </ms> </list> </apply> </apply> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> GridBox </ci> <list> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> </ms> <ms> n </ms> </list> </apply> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ident /> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> <list> <apply> <ci> TagBox </ci> <ms> k </ms> <ident /> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <apply> <ci> Binomial </ci> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Slot </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> </apply> <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> 0 </ms> </list> </apply> <ms> k </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> FractionBox </ci> <ms> p </ms> <ms> q </ms> </apply> <ms> ) </ms> </list> </apply> <ms> j </ms> </apply> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> GridBox </ci> <list> <list> <apply> <ci> TagBox </ci> <ms> k </ms> <ident /> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> <list> <apply> <ci> TagBox </ci> <ms> j </ms> <ident /> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <apply> <ci> Binomial </ci> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Slot </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> &#968; </ms> <ci> PolyGamma </ci> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> - </ms> <ms> j </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> &#950; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> - </ms> <ms> k </ms> </list> </apply> <ci> Zeta </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <apply> <ci> FractionBox </ci> <ms> p </ms> <ms> q </ms> </apply> </list> </apply> <ci> Zeta </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> InterpretTemplate </ci> <lambda> <bvar> <ci> ZetaDump`e1$ </ci> </bvar> <bvar> <ci> ZetaDump`e2$ </ci> </bvar> <apply> <ci> Zeta </ci> <ci> ZetaDump`e1$ </ci> <ci> ZetaDump`e2$ </ci> </apply> </lambda> </apply> </apply> </list> </apply> <ms> - </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> &#950; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> - </ms> <ms> k </ms> </list> </apply> <ci> Zeta </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <apply> <ci> FractionBox </ci> <ms> p </ms> <ms> q </ms> </apply> </list> </apply> <ci> Zeta </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> InterpretTemplate </ci> <lambda> <bvar> <ci> ZetaDump`e1$ </ci> </bvar> <bvar> <ci> ZetaDump`e2$ </ci> </bvar> <apply> <ci> Zeta </ci> <ci> ZetaDump`e1$ </ci> <ci> ZetaDump`e2$ </ci> </apply> </lambda> </apply> </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> TagBox </ci> <ms> &#968; 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</ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> &#8712; </ms> <apply> <ci> SuperscriptBox </ci> <apply> <ci> TagBox </ci> <ms> &#8469; </ms> <apply> <ci> Function </ci> <list /> <integers /> </apply> </apply> <ms> + </ms> </apply> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <ms> p </ms> <ms> &#8712; </ms> <apply> <ci> TagBox </ci> <ms> &#8484; </ms> <apply> <ci> Function </ci> <list /> <integers /> </apply> </apply> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <ms> 0 </ms> <ms> &lt; </ms> <ms> p </ms> <ms> &lt; </ms> <ms> q </ms> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> &#8712; </ms> <apply> <ci> TagBox </ci> <ms> &#8484; </ms> <apply> <ci> Function </ci> <list /> <integers /> </apply> </apply> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> &gt; </ms> <ms> 1 </ms> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2007-05-02





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