Harmonic Geometric Polynomials via Geometric Polynomials and Their Applications

TL;DR

This paper presents explicit representations of harmonic geometric polynomials using geometric polynomials and explores their applications, including identities, integrals, and relations involving Bernoulli numbers and harmonic numbers.

Contribution

It provides new explicit formulas and applications for harmonic geometric polynomials in terms of geometric polynomials, extending previous work.

Findings

Derived explicit representations of harmonic geometric polynomials.

Obtained new integral formulas involving Bernoulli numbers.

Established recurrence relations and evaluated sums involving harmonic numbers.

Abstract

The aim of this study is to show that harmonic geometric polynomials can be represented in terms of geometric polynomials. This problem was first considered by Keller [14]; however, the corresponding coefficients were not fully determined. In the present work, we provide several explicit representations of harmonic geometric polynomials in terms of geometric polynomials. Moreover, several applications of one of these representations are subsequently developed. In particular, we obtain a generalization of the classical identity for the harmonic numbers, compute an integral involving harmonic geometric polynomials and an integral involving products of harmonic geometric and geometric polynomials in terms of Bernoulli numbers. These integral formulas lead to new explicit expressions for Bernoulli numbers. In addition, we give several recurrence relations for harmonic geometric polynomials and evaluate a finite sum involving harmonic numbers and positive powers of integers.