A Curious Binomial Coefficient Sum: $\sum_{j = 0}^{k} \binom{k}{j} \binom{j + n -\ell + 1}{n}$

Question

Let $k, \ell \leq n$ be non-negative integers. Does the following identity simplify? \begin{align} \sum_{j = 0}^{k} \binom{k}{j} \binom{j + n -\ell + 1}{n} = \binom{n - \ell + 1}{n} \phantom1_{2}\mathsf{F}_{1}(-k,n - \ell + 2, 2- \ell; -1) \end{align} where $\!\!\! \phantom1_{2}\mathsf{F}_{1}$ is a hypergeometric function. That is, does the right side have another representation in terms of simple functions given that $k,\ell$ and $n$ are non-negative integers?