The formula for a perhaps basic identity (move from stackexchange) 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!~
 A: I guess the first product is expanded as
$$
\prod_{k=1}^n(1+x+y_k)=(1+x)^n\prod_k(1+y_k(1+x)^{-1})=\sum_{k\geq 0}e_k(y_1,\ldots,y_n)(1+x)^{-k+n}.$$
For the other products you can write
$$
\prod_{j=1}^m\prod_{i=1}^n(1+y_it_j)=\sum_\lambda e_\lambda(y_1,\ldots,y_n)m_\lambda(t_1,\ldots,t_m)=\sum_\lambda e_\lambda(\mathbf{y})m_\lambda(\mathbf{t})$$
where the sum is over $\lambda$ with $\ell(\lambda)\leq m$ and $m_\lambda$ is the monomial symmetric function. It seems that the product you are interested in looks something like
$$
\prod_{j=1}^m\left(\sum_k e_k(y)(1+x_j)^{n-k}\right)=\sum_\lambda e_\lambda(\mathbf{y})m_{\lambda-(n^m)}((\mathbf{1-x})^{-1})
$$
In this formula, $\lambda-(n^m)=(\lambda_1-n,\ldots,\lambda_m-n)$ and you take the obvious definition of $m_\mu$ where $\mu$ is a partition with parts not necessarily positive.
I'm not sure if this is helpful, since I don't know what you want to do with it. Anyway, this is related to a bilinear form on the ring of symmetric functions given by
$$\langle u,v\rangle = (u,\omega(v))$$
where $(\cdot,\cdot)$ is the standard bilinear form having Schur functions as an orthonormal basis and $\omega$ is the automorphism switching the elementary and homogeneous symmetric functions. In particular, I believe this product also can be expanded as
$$\sum_{\lambda} s_{\lambda^t}(\mathbf{y})s_{\lambda-(n^m)}((\mathbf{1-x})^{-1}).$$
