Parametric binomial sums involving harmonic numbers

TL;DR

This paper derives explicit formulas for parametric binomial sums involving harmonic numbers and generalizes relations between Stirling numbers and the zeta function, providing new series for Apéry's constant.

Contribution

It introduces explicit formulas for binomial sums with harmonic numbers and extends the Stirling numbers-zeta relation to polygamma functions, with applications to zeta(3).

Findings

Explicit formulas for sums involving harmonic numbers and binomial coefficients.

Generalization of Stirling numbers and zeta function relation to polygamma functions.

New series representations for Apéry's constant ζ(3).

Abstract

We present explicit formulas for the following family of parametric binomial sums involving harmonic numbers for $p=0,1,2$ and $|t|\leq1$. $$ \sum_{k=1}^{\infty}\frac{H_{k-1}t^k}{k^p\binom{n+k}{k}}\quad \mbox{and}\quad \sum_{k=1}^{\infty}\frac{t^k}{k^p\binom{n+k}{k}}. $$ We also generalize the following relation between the Stirling numbers of the first kind and the Riemann zeta function to polygamma function and give some applications. $$ \zeta(n+1)=\sum_{k=n}^{\infty}\frac{s(k,n)}{kk!}, \quad n=1,2,3,... . $$ As examples, \begin{equation*} \zeta(3)=\frac{1}{7}\sum_{k=1}^{\infty}\frac{H_{k-1}4^k}{k^2\binom{2k}{k}},\quad \mbox{and}\quad \zeta(3)=\frac{8}{7}+\frac{1}{7}\sum_{k=1}^{\infty}\frac{H_{k-1}4^k}{k^2(2k+1)\binom{2k}{k}}, \end{equation*} which are new series representations for the Ap\'{e}ry constant $\zeta(3)$.