Note on the Higher-Order Derivatives of the Hyperharmonic Polynomials and the <em>r</em>-Stirling Polynomials of the First Kind

Abstract

In this paper, we focus on the higher-order derivatives of the hyperharmonic polynomials, which are a generalization of the ordinary harmonic numbers. We determine the hyperharmonic polynomials and their successive derivatives in terms of the r-Stirling polynomials of the first kind and show the relationship between the (exponential) complete Bell polynomials and the r-Stirling numbers of the first kind. Furthermore, we provide a new formula for obtaining the generalized Bernoulli polynomials by exploiting their link with the higher-order derivatives of the hyperharmonic polynomials. In addition, we obtain various identities involving the r-Stirling numbers of the first kind, the Bernoulli numbers and polynomials, the Stirling numbers of the first and second kind, and the harmonic numbers.

Keywords

• hyperharmonic numbers
• hyperharmonic polynomials
• generating function
• <i>r</i>-Stirling polynomials of the first kind
• generalized Bernoulli polynomials
• complete Bell polynomials