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Some integral forms for a generalized zeta function. (English) Zbl 1018.11043

Summary: In this paper some integral forms for a generalized zeta-function are reported. One of them is shown as follows, \[ \zeta(z;a)= \frac{1}{\Gamma(z)} \sum_{k=0}^\infty \int_0^\infty \frac {t^{z-1} e^{-\{a+(1/2)(k+1)k\}t}(1-e^{-(k+1)t})} {1-e^{-t}} dt, \] where \[ \zeta(z;a)= \sum_{m=0}^\infty \frac{1} {(a+m)^z} \qquad (\operatorname{Re} z> 1). \]

MSC:

11M35 Hurwitz and Lerch zeta functions
26A33 Fractional derivatives and integrals
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