I am struggling with a combinatorial sum as apart of a long statistical-mechanics derivation. I would appreciate any help.

I seek the result of the following summation, for integer $\ell,n$, and positive real $z,\alpha$: $$ \sum_{m=0}^{\ell} (-1)^m \displaystyle \frac{\displaystyle\binom{\ell}{z}}{\displaystyle\binom{\alpha m + z}{n}} .$$

For the special case of $\alpha=1$, we would have:

$$ \sum_{m=0}^{\ell} (-1)^m \frac{\displaystyle \binom{\ell}{z}}{\displaystyle \binom{ m + z}{n}}=\displaystyle \frac{n}{(\ell+n)\displaystyle \binom{z+\ell}{n+\ell}} .$$

The latter is manageable, because one can write the binomial coefficient in the denominator in terms of an integral, via $\frac{1}{\binom{n}{k}}=(n+1)\int_0^1 t^k(1-t)^{n-k}dt$, and then exchange summation and integration. This is unduly intricate however, when $\alpha$ is not unity.


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