Some identities on generalized harmonic numbers and generalized harmonic functions

The harmonic numbers and generalized harmonic numbers appear frequently in many diverse areas such as combinatorial problems, many expressions involving special functions in analytic number theory, and analysis of algorithms. The aim of this article is to derive some identities involving generalized harmonic numbers and generalized harmonic functions from the beta functions Fn(x)=B(x+1,n+1),(n=0,1,2,…){F}_{n}\left(x)=B\left(x+1,n+1),\left(n=0,1,2,\ldots ) using elementary methods. For instance, we show that the Hurwitz zeta function ζ(x+1,r)\zeta \left(x+1,r) and r!r\! are expressed in terms of those numbers and functions, for every r=2,3,4,5r=2,3,4,5.