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!~



The formula for a perhaps basic identity (move from stackexchange)
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
$$\sum\_{\lambda} s\_{\lambda^t}(\mathbf{y})s\_{\lambda-(n^m)}((\mathbf{1-x})^{-1}).$$


