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?

• $\binom{n-l+1}{n}=0$ if $l > 1$ I guess – GEdgar Apr 17 '12 at 19:27
• Superficially it appears that the right side vanishes, but you must also consider the hypergeometric function. The sum is equal to $2^{k-1} k$ if $l = 2$ and $n = 1$. – user02138 Apr 17 '12 at 19:46
• What do you consider to be a "simple function"? The theory behind Gosper's algorithm will tell you that if the sum exists as a hypergeometric then it's a rational polynomial times the summand. – Peter Taylor Apr 17 '12 at 20:44
• Following @GEdgar's comment: how do you define $\binom{a}{b}$ when $a < b$ -- not as $0$? – Clement C. Jul 18 '16 at 17:20
• Are you interested in simplifying the left hand side? At least in a formal form? – rrogers Apr 22 '17 at 13:22

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.