SOME SUMS VIA EULER'S TRANSFORM

ÖMÜR N., KOPARAL S., Elkhiri L.

HONAM MATHEMATICAL JOURNAL, no.3, pp.365-377, 2024 (ESCI) identifier

  • Publication Type: Article / Article
  • Publication Date: 2024
  • Doi Number: 10.5831/hmj.2024.46.3.365
  • Journal Name: HONAM MATHEMATICAL JOURNAL
  • Journal Indexes: Emerging Sources Citation Index (ESCI)
  • Page Numbers: pp.365-377
  • Bursa Uludag University Affiliated: Yes

Abstract

In this paper, we give some sums involving the generalized harmonic numbers Hrn(sigma) and the (q,r)-binomial coefficient (Lk)q,r by using Euler's transform. For example, for (c,r)is an element of Z+x R+, & sum;(infinity)(n=0)& sum;(n)(k=0)(-1)(k)(n+r n-k)c(n+1)H(k)(r-1)(sigma)/(n+1)(1+c)(n+1)=-(c+1/sigma)ln(1+c sigma)+c, and & sum;(n)(k=0)(nk)(Lk)(2,r)=& sum;(n)(j=0)& sum;(j)(k=0)(-1)(k)(j-k+2L+rj-k)(r n-j)(Lk)(2), where sigma is appropriate parameter, Hrn(sigma) is the generalized hyperhar-monic number of order r and (Lk)q is the q-binomial coefficient.