On the values of the Dirichlet series associated with certain hyperharmonic numbers
Ramanujan Journal, vol.69, no.1, 2026 (SCI-Expanded, Scopus)
- Publication Type: Article / Article
- Volume: 69 Issue: 1
- Publication Date: 2026
- Doi Number: 10.1007/s11139-025-01270-9
- Journal Name: Ramanujan Journal
- Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH
- Keywords: Dirichlet series, Euler sum, Harmonic number, Hyperharmonic number, Riemann zeta values
- Akdeniz University Affiliated: Yes
Abstract
Let n and r be natural numbers and (Formula presented.) be the hyperharmonic extension of the odd harmonic numbers On=1+1/3+1/5+⋯+1/2n-1. For this extension, we obtain a generating function, recursion relations, and closed-form evaluation formulas in terms of hyperharmonic numbers. Moreover, we show that the Euler-type sums (Formula presented.) can be written in terms of zeta values and log-sine integrals. Here fn∈on(r), -1n-1h~nr,h2nr,h~2nr, and hnr and h~nr stand for the hyperharmonic and skew-hyperharmonic numbers, respectively. We further present evaluation formulas for the nonlinear Euler sums whose summands involve the variant of harmonic numbers, hyperharmonic number hqn+1 and reciprocal binomial coefficients.