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)) /; Element[n, Integers] && n > 0 && Element[p, Integers] && 0 < p < q && Element[q, Integers] && q > 1

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</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> <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> <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> <mtext> </mtext> <mo> + </mo> <mrow> <mfrac> <mrow> <mtext> </mtext> <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> 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</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> 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2007-05-02