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Can the following expression be reduced to some more simplified expression?

$\sum_{j=0}^{k} \binom{n-a + m-j}{n-a} \times \binom{a-1 + j}{a-1}$

n,m and a are some positive constants with a < n and k < m.

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1 Answer

Maple 16 writes it in terms of hypergeometric functions: $${n-a+m\choose n-a}{\mbox{$_2$F$_1$}(a,-m;\,-n+a-m;\,1)}-{n-a+m-k-1 \choose n-a}{a+k\choose a-1} {\mbox{$_3$F$_2$}(1,k+1+a,-m+1+k;\,k+2,-n+a-m+k+1;\,1)}$$

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