Asymptotic expansions for the alternating Hurwitz zeta function and its derivatives

• Published in: Journal of mathematical analysis and applications Vol. 537; no. 1; p. 128306
• Main Authors: Hu, Su, Kim, Min-Soo
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.09.2024
• Subjects: Alternating Hurwitz zeta functions; Asymptotic expansions; Error bounds; Euler polynomials; Gamma function; Error bounds; Asymptotic expansions; Gamma function; Euler polynomials; Alternating Hurwitz zeta functions
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2024.128306

LetζE(s,q)=∑n=0∞(−1)n(n+q)s be the alternating Hurwitz (or Hurwitz-type Euler) zeta function. In this paper, we obtain the following asymptotic expansion of ζE(s,q)ζE(s,q)∼12q−s+14sq−s−1−12q−s∑k=1∞E2k+1(0)(2k+1)!(s)2k+1q2k+1, as |q|→∞, where E2k+1(0) are the special values of odd-order Euler polynomials at 0, and we also consider representations and bounds for the remainder of the above asymptotic expansion. In addition, we derive the asymptotic expansions for the higher order derivatives of ζE(s,q) with respect to its first argumentζE(m)(s,q)≡∂m∂smζE(s,q), as |q|→∞. Finally, we also prove a new exact series representation of ζE(s,q).