New sums mixing harmonic numbers and central binomial coefficients
Micheal Bataille, Robert Frontczak
TL;DR
This work extends the study of sums with inverse central binomial coefficients by incorporating harmonic numbers. By differentiating known central-binomial identities with respect to a parameter and developing Abel-type transformations, the authors derive closed forms for harmonic-number–weighted sums, establish recurrences for the power-sum families $U_d(n)$ and $V_d(n)$, and introduce a second harmonic-sum family with a distinct weighting. They further obtain a final identity involving $H_k^2$ and $H_k^{(2)}$, linking to Parker’s formula and connecting to prior results by Bataille, Frontczak, and others. The results yield explicit, computable expressions and multiple special cases, and point to future exploration of related polynomials $P_d(n)$ and $Q_d(n)$ in this combinatorial-harmonic setting.
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.