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Category Archives: PolyGamma
Integral representation of generalized Euler sums
$$\sum_{k=1}^\infty\frac{H^{(p)}_k}{k^q} = \zeta(p)\zeta(q) +(1)^{p}\frac{1}{ (p1)!}\int^1_0\frac{\mathrm{Li}_q(x)\log(x)^{p1}}{1x}\,dx$$ $$\textit{proof}$$ Note that $$\psi_0(a+1)= \int^1_0\frac{1x^a}{1x}\,dx$$ By differentiating with respect to \(a\) , \(p \) times we have $$\psi_p(a+1) = \frac{\partial}{\partial a^p}\int^1_0\frac{1x^a}{1x}\,dx$$ $$\psi_p(a+1) = \int^1_0\frac{x^a\log(x)^{p}}{1x}\,dx$$ Let \( a =k\) $$\psi_{p1}(k+1) = \int^1_0\frac{x^k\log(x)^{p1}}{1x}\,dx$$ Use the relation to polygamma $$H^{(p)}_k … Continue reading
Posted in Euler sum, PolyGamma, Polylogarithm
Tagged Euler, Integral, polylgoarithm, proof, representation, sum
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Relation between polygamma and Hurwitz zeta function proof
\( \forall \,\, n\geq 1 \) $$\psi_{n}(z) \, = \, (1)^{n+1}n!\,\zeta(n+1,z)$$ $$\textit{proof}$$ Use the series representation of the digamma $$\psi_{0}(z) = \gamma\frac{1}{z}+ \sum_{n=1}^\infty\frac{z}{n(n+z)}$$ This can be written as the following $$\psi_{0}(z) = \gamma + \sum_{k=0}^\infty\frac{1}{k+1}\frac{1}{k+z}$$ By differentiating with respect to … Continue reading