The following question is moved from math stackexchange. It seems that this is not a popular question, but I really want to know the answer so I moved it to here. The question reads as follows.

We know the expansion of the following product

$\displaystyle\prod_{k=1}^n(1+x+y_k)$

can be expressed by the formula

$\displaystyle\sum_{k=0}^n(1+x)^{n-k}e_k(y_1, \ldots, y_n),$

where the $e_k$'s are the elementary symmetric functions,

$e_k (x_1 , \ldots , x_n )=\displaystyle\sum_{1\le j_1 < j_2 < \dots < j_k \le n} x_{j_1} \dotsm x_{j_k}.$

My question is whether we have a nice formula for the expansion of the following product

$\displaystyle\prod_{1\leq k\leq n, 1\leq\ell\leq m}(1+x_\ell+y_k).$

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!~

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