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

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?

Answer

Let us denote:
$$\begin{equation} S_k^{n,l} := \sum\limits_{j=0}^k \left( \begin{array}{c} k \\ j \end{array} \right) \left( \begin{array}{c} n-l+1+j \\ n \end{array} \right) \end{equation}$$
Then we have:
$$\begin{eqnarray} S_k^{n,l} &:=& \left. \frac{d^n}{n! d x^n} x^{n-l+1} \cdot \left(1+x\right)^k \right|_{x=1} \\ &=& \frac{1}{n!} \sum\limits_{p=0}^n \left(\begin{array}{c} n \\ p \end{array} \right) (n-l+1)_{(p)} k_{(n-p)} 2^{k-n+p} \\ &=&2^{k-n} \left( \begin{array}{c} k \\ n \end{array} \right) {}_2F_1\left[4-n,-n,1+k-n;2\right] \end{eqnarray}$$

The last result, that in terms of the hypergeometric function is not particularily useful when $n$, $l$, $k$ are integers. However the result above is useful for example if $k$ is large.