On some Binomial Coefficient Identities with Applications

Abstract

We present a different proof of the following identity due to Munarini, which generalizes a curious binomial identity of Simons. \begin{align*} \sum_{k=0}^{n}\binom{\alpha}{n-k}\binom{\beta+k}{k}x^k &=\sum_{k=0}^{n}(-1)^{n+k}\binom{\beta-\alpha+n}{n-k}\binom{\beta+k}{k}(x+1)^k, \end{align*} where $n$ is a non-negative integer and $\alpha$ and $\beta$ are complex numbers, which are not negative integers. Our approach is based on a particularly interesting combination of the Taylor theorem and the Wilf-Zeilberger algorithm. We also generalize a combinatorial identity due to Alzer and Kouba, and offer a new binomial sum identity. Furthermore, as applications, we give many harmonic number sum identities. As examples, we prove that \begin{equation*} H_n=\frac{1}{2}\sum_{k=1}^{n}(-1)^{n+k}\binom{n}{k}\binom{n+k}{k}H_k \end{equation*} and \begin{align*} \sum_{k=0}^{n}\binom{n}{k}^2H_kH_{n-k}=\binom{2n}{n} \left((H_{2n}-2H_n)^2+H_{n}^{(2)}-H_{2n}^{(2)}\right). \end{align*}