Parametric binomial sums involving harmonic numbers

Abstract

We present explicit formulas for the following family of parametric binomial sums involving harmonic numbers for $p=0,1,2$ and $|t|\le 1$.

$$\begin{aligned} \sum _{k=1}^{\infty }\frac{H_{k-1}t^k}{k^p\left( {\begin{array}{c}n+k\\ k\end{array}}\right) } \quad \text{ and }\quad \sum _{k=1}^{\infty }\frac{t^k}{k^p\left( {\begin{array}{c}n+k\\ k\end{array}}\right) }. \end{aligned}$$

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.

$$\begin{aligned} \zeta (n+1)=\sum _{k=n}^{\infty }\frac{s(k,n)}{kk!}, \quad n=1,2,3,\ldots . \end{aligned}$$

As examples,

$$\begin{aligned} \zeta (3)=\frac{1}{7}\sum _{k=1}^{\infty }\frac{H_{k-1}4^k}{k^2\left( {\begin{array}{c}2k\\ k\end{array}}\right) }, \quad \text{ and }\quad \zeta (3)=\frac{8}{7}+\frac{1}{7}\sum _{k=1}^{\infty } \frac{H_{k-1}4^k}{k^2(2k+1)\left( {\begin{array}{c}2k\\ k\end{array}}\right) }, \end{aligned}$$

which are new series representations for the Apéry constant $\zeta (3)$.