Stieltjes constants appearing in the Laurent expansion of the hyperharmonic zeta function

CAN M., DİL A., KARGIN L.

Ramanujan Journal, vol.61, no.3, pp.873-894, 2023 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Volume: 61 Issue: 3
  • Publication Date: 2023
  • Doi Number: 10.1007/s11139-022-00676-z
  • Journal Name: Ramanujan Journal
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH
  • Page Numbers: pp.873-894
  • Keywords: Euler sum, Harmonic numbers, Hyperharmonic numbers, Integro-exponential function, Laurent expansion, Stieltjes constant, Zeta values
  • Akdeniz University Affiliated: Yes

Abstract

In this paper, we consider meromorphic extension of the function ζh(r)(s)=∑k=1∞hk(r)ks,Re(s)>r(which we call hyperharmonic zeta function) where hn(r) are the hyperharmonic numbers. We establish certain constants, denoted γh(r)(m), which naturally occur in the Laurent expansion of ζh(r)(s). Moreover, we show that the constants γh(r)(m) and integrals involving the generalized exponential integral can be written as a finite combination of some special constants.