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I am beginning a course in combinatorics and I'd like to know if there is a formula to compute:

$$\sum_{k=0}^r \binom{m}{k} \binom{n}{r-k}x^{k},$$

where $r \leq min(m,n)$ are integers and $x$ is an indeterminate. I know that the Vandermonde's formula reads:

$$\sum_{k=0}^{r} \binom{m}{k} \binom{n}{r-k} = \binom{m+n}{r}.$$

But I don't see an easy way to compute the sum I want using this Vandermonde's formula.

Thanks for your help!

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1 Answer 1

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You could call it $${n\choose r}{\mbox{$_2$F$_1$}(-r,-m;\; n-r+1;\,x)}$$

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  • $\begingroup$ OK, thanks for your answer. But I don't know what are $F_1$ and $\binom{n}{r}_2$. Is there a way to compute this sum using standard functions and/or algebraic integrals? $\endgroup$
    – Libli
    Dec 21, 2017 at 21:14
  • $\begingroup$ ${}_2F_1$ is a hypergeometric function. $\endgroup$ Dec 21, 2017 at 22:51

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