Some sums via euler's transform

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.