New sums mixing harmonic numbers and central binomial coefficients

TL;DR

This paper introduces new classes of sums involving inverse binomial coefficients and harmonic numbers, providing recursive solutions for power sums with these components, advancing the understanding of their mathematical properties.

Contribution

The paper presents novel sums combining harmonic numbers and central binomial coefficients, along with recursive formulas for related power sums, which were not previously documented.

Findings

Derived recursive solutions for sums involving harmonic numbers and binomial coefficients

Established new classes of sums with inverse binomial coefficients and harmonic numbers

Enhanced understanding of power sums with these mathematical components

Abstract

We study two new classes of sums with inverse binomial coefficients and harmonic numbers. In addition we establish recursive solutions to the following power sums \begin{equation*} U_d(n) = \sum_{k=1}^n \frac{2^{2k}}{\binom{2k}{k}} \cdot k^d \quad \mbox{and}\quad V_d(n) = \sum_{k=1}^n \frac{2^{2k}}{\binom{2k}{k}}\cdot k^d\,H_k, \end{equation*} where $d$ is a positive integer.