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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?

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    $\begingroup$ $\binom{n-l+1}{n}=0$ if $l > 1$ I guess $\endgroup$ – GEdgar Apr 17 '12 at 19:27
  • $\begingroup$ 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 $n = 1$. $\endgroup$ – user02138 Apr 17 '12 at 19:46
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    $\begingroup$ 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. $\endgroup$ – Peter Taylor Apr 17 '12 at 20:44
  • $\begingroup$ Following @GEdgar's comment: how do you define $\binom{a}{b}$ when $a < b$ -- not as $0$? $\endgroup$ – Clement C. Jul 18 '16 at 17:20
  • $\begingroup$ Are you interested in simplifying the left hand side? At least in a formal form? $\endgroup$ – rrogers Apr 22 '17 at 13:22
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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.

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