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Sums involving the Hurwitz zeta function. (English) Zbl 0989.11043

Let as usual \(\zeta(s,\alpha)\) be the Hurwitz zeta-function and \(\Gamma(s)\) be the Euler gamma function. Let \(\operatorname{Re}\alpha>0\), \(\operatorname{Re} p>0\), \(\operatorname{Re} q>0\) and \(|z|<|\alpha|\). In the paper, the authors obtain the following equality: \[ \int_0^1 u^{p-1}(1-u)^{q-1}\{\zeta(s, \alpha-zu)-\zeta(s, \alpha)\} du=\sum_{n=1}^\infty{\Gamma(s+n)\Gamma(p+n)\Gamma(q) \over \Gamma(s)n!\Gamma(p+q+n)}\zeta(s+n,\alpha)z^n. \] From this they derive a closed formula in which an infinite series of the type \(\sum_{m=2}^\infty {z^m\over m+\lambda}\zeta(m,\alpha)\) is expressed by finite sums involving Stirling numbers and multiple gamma functions. Next, the authors consider an explicit expression related to \(\zeta'(-k, \alpha)\), \(k\) is a non negative integer; to the constants \(R_r:=\lim_{\alpha\to 0}\alpha\exp {\delta\over\delta s}\zeta_r(0, \alpha)\), where \(\zeta_r(s, \alpha)\) is the \(r\)th Hurwitz zeta-function; and to the logarithmic derivative of the multiple gamma function.

MSC:

11M35 Hurwitz and Lerch zeta functions
33B15 Gamma, beta and polygamma functions
11M41 Other Dirichlet series and zeta functions
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