Some Identities Involving the Generalized Harmonic Numbers

Abstract

In this paper, we discuss the properties of a class of generalized harmonic numbers $H_{n,r}$. Using Riordan arrays and generating functions, we establish some identities involving $H_{n,r}$. Furthermore, we investigate certain sums related to harmonic polynomials $H_n(z)$. In particular, using the Riordan array method, we explore interesting relationships between these polynomials, the generating Stirling polynomials, the Bernoulli polynomials, and the Cauchy polynomials. Finally, we obtain the asymptotic expansion of certain sums involving $H_{n,r}$.