Daehee, hyperharmonic, and power sums polynomials

Abstract

In this paper we consider the Daehee numbers and polynomials of the first and second kind, and give several explicit representations for them. In particular, we express the Daehee polynomials as the derivative of a generalized binomial coefficient. This is done by performing the Stirling transform of the power sum polynomial S k (x) associated with the sum of kth powers of the first n positive integersFurthermore, we show the relationship between the Daehee polynomials and the hyperharmonic polynomials. This allows us to also express the hyperharmonic polynomials as the derivative of a generalized binomial coefficient. In addition, we reformulate and generalize a number of identities involving the Stirling, Bernoulli, and harmonic numbers, in terms of the Daehee and hyperharmonic polynomials. Finally, in the light of a recent result of Kargın et al., we discuss a further extension of the obtained identities in terms of the r-Stirling numbers.

A note on the hyper-sums of powers of integers, hyperharmonic polynomials and r-Stirling numbers of the first kind

“…In this section we will provide further identities involving the r-Stirling numbers of the first kind, the Bernoulli numbers and polynomials, the ordinary Stirling numbers of the first and second kinds, and the harmonic numbers. Firstly, from [5,Equation (69)], it follows that, for k ≥ 0, the Bernoulli polynomials B k (x) can be expressed in terms of H (x) j+1 as…”