We know the expansion of the following product
$\prod_{k=1}^n(1+x+y_k)$
can be expressed by the formula
$\sum_{k=0}^n(1+x)^{n-k}s_k(y_1, \ldots, y_n),$
where the $s_k$'s are the elementary symmetric functions. My question is whether we have a nice formula for the expansion of the following product
$\prod_{1\leq k\leq n, 1\leq\ell\leq m}(1+x_\ell+y_k).$
Reference for the nice formula of the above expression will be highly appreciated. (It seems to me that it is related to generating functions, but I have no background in combinatorics.)
Thanks!~