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

Nishimoto, Katsuyuki; Yen, Chen-E.; Lin, Ming-Lai

J. Fractional Calc. 22, 91-97 (2002).

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

Keywords:

integral representations; generalized zeta-function