An integral form for a generalized zeta function. (English) Zbl 1018.11044

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

J. Fractional Calc. 22, 99-102 (2002).

Summary: We extend Theorem 4 of the preceding paper [ibid. 22, 91-97 (2002; Zbl 1018.11043)] as follows: \[ \zeta(z;a)= \sum_{i=0}^{k-1} \frac{1}{\Gamma(z)} \int_0^\infty \frac {t^{z-1} e^{-(a+i)t}} {1-e^{-kt}} dt \qquad (\operatorname{Re} z> 1), \] where \(\zeta(z;a)= \sum_{m=0}^\infty \frac{1} {(a+m)^z}\) \((\operatorname{Re} z>1)\), and \(a>0\) and \(k>1\) is a fixed positive integer.

MSC:

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

Citations:

Zbl 1018.11043