Harmonic Geometric Polynomials via Geometric Polynomials and Their Applications

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.