On the values of the Dirichlet series associated with certain hyperharmonic numbers

Kara M., CİCİMEN M., Mutluer M., Aytaç P.

Ramanujan Journal, cilt.69, sa.1, 2026 (SCI-Expanded, Scopus) identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 69 Sayı: 1
  • Basım Tarihi: 2026
  • Doi Numarası: 10.1007/s11139-025-01270-9
  • Dergi Adı: Ramanujan Journal
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH
  • Anahtar Kelimeler: Dirichlet series, Euler sum, Harmonic number, Hyperharmonic number, Riemann zeta values
  • Akdeniz Üniversitesi Adresli: Evet

Özet

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.