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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!~

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  • $\begingroup$ Hint: This is symmetric in the $x_i$ and the $y_j$. $\endgroup$ – Martin Brandenburg Nov 1 '14 at 9:28
  • $\begingroup$ Um...yes...i know...and...I do expect the formula is also symmetric in the $x_i$ and the $y_j$. Is it some related to elementary symmetric functions again or something else? Please give me more hint... $\endgroup$ – GRR Nov 1 '14 at 10:25
  • $\begingroup$ Also posted to MO, mathoverflow.net/questions/185937/… $\endgroup$ – Gerry Myerson Nov 1 '14 at 22:27

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