Asymptotic expansions for the alternating Hurwitz zeta function and its derivatives. (English) Zbl 07837281
The alternating Hurwitz (or Hurwitz-type Euler) zeta function \(\zeta_E (s, q)\) is defined by
\[
\zeta_E (s, q) = \sum_{n = 0}^\infty \frac{(- 1)^n}{(n + q)^s},
\]
for \(\operatorname{Re} (s)>0\) and \(q\neq 0, -1, -2, \ldots\) One can see that \(\zeta_E (s, q)\) can be analytically continued to the entire \(s\)-plane without any pole and it satisfies the following identities:
\[
\zeta_E (s, q+1) + \zeta_E (s, q)= q^{-s}
\]
and
\[
\frac{\partial}{\partial q}\zeta_E (s, q)=-s \zeta_E (s+1, q).
\]
In this paper under review, the authors are inspired from the work of G. Nemes [Proc. R. Soc. Lond., A, Math. Phys. Eng. Sci. 473, No. 2203, Article ID 20170363, 16 p. (2017; Zbl 1404.33018)] and they first obtain the following asymptotic expansion of \(\zeta_E (s, q)\):
\[
\zeta_E (s, q) \sim \frac{1}{2} q^{- s} + \frac{1}{4} sq^{- s - 1} - \frac{1}{2} q^{-s} \sum_{k = 1}^\infty \frac{E_{2 k + 1} (0)}{(2 k + 1) !} \frac{(s)_{2 k + 1}}{q^{2 k + 1}},
\]
as \(|q|\to \infty\) in the sector \(|\arg q|\leq \pi-\delta\), with \(s\) and \(\delta>0\) being fixed, where \(E_{2k+1}(0)\) are the special values of odd-order Euler polynomials at \(0\).
They also derive the asymptotic expansions for the higher order derivatives of \(\zeta_E(s,q)\) with respect to its first argument \[ \zeta_E^{(m)}(s, q) \equiv \frac{\partial^m}{\partial s^m} \zeta_E(s, q), \] as \(|q|\to \infty\) in the sector \(|\arg q|\leq \pi-\delta\), with \(s\) and \(\delta>0\) being fixed (see Theorems 3.9 and 3.14). Finally, they prove a new exact series representation of the convergent expansion of \(\zeta_E(s, q)\). Notice that a similar representation was originally established by G. Nemes [Proc. R. Soc. Lond., A, Math. Phys. Eng. Sci. 473, No. 2203, Article ID 20170363, 16 p. (2017; Zbl 1404.33018)] for the Hurwitz zeta function \(\zeta(s, q)\).
They also derive the asymptotic expansions for the higher order derivatives of \(\zeta_E(s,q)\) with respect to its first argument \[ \zeta_E^{(m)}(s, q) \equiv \frac{\partial^m}{\partial s^m} \zeta_E(s, q), \] as \(|q|\to \infty\) in the sector \(|\arg q|\leq \pi-\delta\), with \(s\) and \(\delta>0\) being fixed (see Theorems 3.9 and 3.14). Finally, they prove a new exact series representation of the convergent expansion of \(\zeta_E(s, q)\). Notice that a similar representation was originally established by G. Nemes [Proc. R. Soc. Lond., A, Math. Phys. Eng. Sci. 473, No. 2203, Article ID 20170363, 16 p. (2017; Zbl 1404.33018)] for the Hurwitz zeta function \(\zeta(s, q)\).
Reviewer: Sami Omar (Sukhair)
MSC:
| 11M35 | Hurwitz and Lerch zeta functions |
| 11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |
| 33B15 | Gamma, beta and polygamma functions |
Keywords:
asymptotic expansions; alternating Hurwitz zeta functions; error bounds; Euler polynomials; gamma functionCitations:
Zbl 1404.33018Software:
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