Taiwanese Journal of Mathematics

The Multiple Hurwitz Zeta Function and the Multiple Hurwitz-Euler Eta Function

Junesang Choi and H. M. Srivastava

Full-text: Open access

Abstract

Almost eleven decades ago, Barnes introduced and made a systematic investigation on the multiple Gamma functions $\Gamma_n$. In about the middle of 1980s, these multiple Gamma functions were revived in the study of the determinants of Laplacians on the $n$-dimensional unit sphere ${\bf S}^n$ by using the multiple Hurwitz zeta functions $\zeta_n(s,a)$. In this paper, we first aim at presenting a generalized Hurwitz formula for $\zeta_n(s,a)$ together with its various special cases. Secondly, we give analytic continuations of multiple Hurwitz-Euler eta function $\eta_n(s,a)$ in two different ways. As a by-product of our second investigation, a relationship between $\eta_n(-\ell,a)$ $(\ell \in \mathbb{N}_0)$ and the generalized Euler polynomials $E_\ell^{(n)}(n-a)$ is also presented.

Article information

Source
Taiwanese J. Math., Volume 15, Number 2 (2011), 501-522.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500406218

Digital Object Identifier
doi:10.11650/twjm/1500406218

Mathematical Reviews number (MathSciNet)
MR2810165

Zentralblatt MATH identifier
1273.11133

Subjects
Primary: 11M06: $\zeta (s)$ and $L(s, \chi)$ 11M99: None of the above, but in this section
Secondary: 33B99: None of the above, but in this section

Keywords
Gamma function multiple Gamma functions Riemann zeta function Hurwitz zeta function multiple Hurwitz zeta function eta function multiple Hurwitz-Euler eta function Taylor-Maclaurin series expansion generalized Bernoulli numbers and polynomials

Citation

Choi, Junesang; Srivastava, H. M. The Multiple Hurwitz Zeta Function and the Multiple Hurwitz-Euler Eta Function. Taiwanese J. Math. 15 (2011), no. 2, 501--522. doi:10.11650/twjm/1500406218. https://projecteuclid.org/euclid.twjm/1500406218


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