PolyGamma[-2 n, 1/4] == (1/(-1 + 2 n)!) ((-1)^n 2^(1 - 6 n) Pi^(1 - 2 n) PolyGamma[-1 + 2 n, 1/4] - ((-2 + 4^n) Derivative[1][Zeta][1 - 2 n])/16^n + ((4^(-1 - 2 n) ((-4^n + 16^n) Pi + (-4 + 4^n) Log[4]))/n) BernoulliB[2 n] + (1/n) (2^(-1 - 4 n) (-EulerGamma + I Pi - 2 I n Pi + n Log[16 Pi^8] + 8 n (EulerGamma + PolyGamma[2 n]))) - 4^(1 - 2 n) Sum[(-4)^k Binomial[-1 + 2 n, k] (Sum[(Binomial[k, j] PolyGamma[1 - j + k] Zeta[j - k, 3/4])/4^j, {j, 0, k}] - PolyGamma[1 + k] Zeta[-k] - Derivative[1][Zeta][-k]), {k, 0, -1 + 2 n}] - 4^(1 - 2 n) Sum[((-1)^j Binomial[-1 + 2 n, j])/ (-1 - j + 2 n), {j, 0, -2 + 2 n}] - 4^(1 - 2 n) Sum[(-1)^j Binomial[-1 + 2 n, j] Sum[(Binomial[-1 - j + 2 n, k] k! PolyLog[1 + k, I])/(-((Pi I)/2))^k, {k, 0, -1 - j + 2 n}], {j, 0, -2 + 2 n}] - (2 I Pi)^(1 - 2 n) Sum[((I Pi)/2)^j Binomial[-1 + 2 n, j] (-1 - j + 2 n)! Zeta[-j + 2 n], {j, 0, -2 + 2 n}]) /; Element[n, Integers] && n > 0

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type='integer'> -1 </cn> <ci> j </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Zeta </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <pi /> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 4 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <pi /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='integer'> 4 </cn> <ci> n </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 16 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> -4 </cn> <apply> <power /> <cn type='integer'> 4 </cn> <ci> n </ci> </apply> </apply> <apply> <ln /> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> BernoulliB </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -4 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <eulergamma /> </apply> <apply> <times /> <imaginaryi /> <pi /> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> n </ci> <pi /> </apply> </apply> <apply> <times /> <ci> n 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<apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Binomial </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> <ci> j </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> 4 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> -2 </cn> </apply> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <ci> Binomial </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> <ci> j </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <ci> 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<apply> <power /> <cn type='integer'> -4 </cn> <ci> k </ci> </apply> <apply> <ci> Binomial </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> <ci> k </ci> </apply> <apply> <plus /> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> <apply> <ci> Binomial </ci> <ci> k </ci> <ci> j </ci> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Zeta </ci> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <cn type='rational'> 3 <sep /> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn 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2007-05-02