# A Curious Binomial Coefficient Sum

Let $k, l \leq n$ be non-negative integers. Does the following identity simplify? \begin{align} \sum_{j = 0}^{k} \binom{k}{j} \binom{j + n -l + 1}{n} = \binom{n - l + 1}{n} \phantom1_{2}\mathsf{F}_{1}(-k,n - l + 2, 2- l; -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 $k,l$ and $n$ are non-negative integers?

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$\binom{n-l+1}{n}=0$ 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