Expressing symmetric function in power-sum basis

I am trying to prove the following identity $$$$\prod_{i=1}^{m}(1-x_{i}z)^{-u}\prod_{j=1}^{n}(1-y_{i}z)^{-v} \prod_{i=1}^{m}\prod_{j=1}^{n}(1-(x_i +y_j)z)^{-w}\\ = \sum_{\lambda, \mu}c_{\lambda, \mu, \nu, \eta}(u,v,w)\frac{s_{\lambda}({\bf{x}})s_{\mu}({\bf{y}})}{h_{\lambda}h_{\mu}}z^{\mid \mu \mid+ \mid \lambda \mid} \tag{*}\label{star}$$$$ where $$c_{\lambda, \mu, \nu, \eta}(u,v,w)= \sum_{\nu\subset \lambda, \,\eta\subset\mu\\ \lvert \nu\rvert= \lvert \eta \rvert} c_{\lambda, \mu}^{\nu, \eta}d_{\nu, \eta}P_{\lambda / \nu , \mu / \eta}(u,v,w,m,n)$$ where $$c_{\lambda, \mu}^{\nu, \eta}\in \mathbb{Q}$$ and $$d_{\nu, \eta}\in \mathbb{Q}[w]$$ is a monic polynomial of degree $$\lvert \nu \rvert$$ and $$P_{\lambda / \nu , \mu / \eta}= \prod_{(i,j)\in \lambda / \nu}(u+nw +j-i)\prod_{(i,j)\in \mu / \eta}(v+mw+j-i)$$, $$h_{\lambda}$$ denoting the hook length.

For the case $$m=n=1$$ we can prove this formula, but in general I can't see the proof. Initially I am trying the following approach, as the LHS of \eqref{star} has an explicit form and we know

$$\ln\Big(\prod_{i\geq 1}\frac{1}{1-x_i z}\Big) = \sum_{n\geq 1}p_n ({\bf{x}})\frac{z^n}{n}$$ where $${\mathbf{x}}:= (x_1, x_2, \dotsc)$$, where $$p_{n}({\mathbf{x}})$$ denotes the power symmetric basis. We express the LHS in the power symmetric basis and evaluating to $$x_i = 0$$ for $$i> m$$ and $$y_j =0$$ for $$j> n$$.

Similarly we can write

$$\ln\Bigl(\prod_{i\geq 1}\prod_{j\geq 1}\frac{1}{1-(x_i +y_j) z}\Bigr) = \sum_{j\geq 1} \sum_{n\geq 1}p_n ({\mathbf{x}}+y_j)\frac{z^n}{n}$$ where $${\mathbf{x}}+y_j = (x_1 + y_j , x_2 +y_j,\dotsc)$$ in power symmetric basis.

Now for the RHS we can we write it as a power symmetric basis? Cauchy identity comes to mind but this expression is more general. Any idea will be really helpful.

• It's possible that this question is related to Problem 86 at math.mit.edu/~rstan/ec/ch7supp.pdf (solution at math.mit.edu/~rstan/ec/ch7suppsol.pdf). We should generalize the definition of $\vartheta$ to $\vartheta(p_k)=\sum_{i=0}^k {k\choose i}p_i(x)p_{k-i}(y)=p_k(x+y)$, where $x+y$ denotes the variables $x_i+y_j$. Dec 5, 2021 at 19:44
• Can you please explain how $\vartheta$ helps in this case?
– GGT
Dec 6, 2021 at 12:29
• I don't have a proof, only a feeling that Problem 86 is related. This is because (1) is involves evaluation at $x_1+1,\dots,x_n+1$ while your question involves evaluation at $x_i+y_j$, and (2) my problem involves $\prod_{u\in\lambda/\mu}(t+c(u))$, which is similar to products appearing in your conjecture. Dec 6, 2021 at 22:23
• I see it now thanks
– GGT
Dec 7, 2021 at 0:38
• Should $\mathbf x + y_j$ really be $(x_1 + y_j, x_2 + y_i, \dotsc)$, or should the second entry be $x_2 + y_j$? \\ Also, TeX note: absolute-value-type operations are better typeset as $\lvert\nu\rvert$ \lvert\nu\rvert, not $\mid\nu\mid$ \mid\nu\mid. (Note the difference between, for example, $2\lvert\nu\rvert$ 2\lvert\nu\rvert and $2\mid\nu\mid$ 2\mid\nu\mid.) I have edited accordingly. Apr 6, 2022 at 23:16

This is not an answer in full, I was hoping if this initiate some more discussion. My idea is to write both sides in a power symmetric basis and show the equality comparing the general coefficient. So we have the Cauchy identities, $$\sum_{\lambda}s_{\lambda}({\bf{x}})s_{\lambda}({\bf{y}}) =\exp\left(\sum_{k\geq 1} \frac{p_{k}({\bf{x}})}{k} p_{k}({\bf{y}})\right)$$ Now if we do specialisation replacing $$y_j= s$$ we have $$\sum_{\lambda}s_{\lambda}({\bf{x}})s_{\lambda}(s,s,s,\ldots) =\exp\left(\sum_{k\geq 1} \frac{p_{k}({\bf{x}})}{k} p_{k}(s,s,\ldots)\right)$$ By the product content formula
we have $$\sum_{\lambda}s_{\lambda}({\bf{x}})\prod_{(i,j)\in \lambda}\frac{(z+i-j)}{h_{\lambda}((i,j))} =\exp\left(\sum_{k\geq 1} \frac{p_{k}({\bf{x}})}{k} m z^k\right)\tag{*}$$ Similarly we can have $$\sum_{\mu}s_{\mu}({\bf{y}})\prod_{(i,j)\in \lambda}\frac{(z+i-j)}{h_{\mu}((i,j))} =\exp\left(\sum_{k\geq 1} \frac{p_{k}({\bf{y}})}{k} n z^k\right)\tag{**}$$
Product of $$*$$ and $$**$$ gives expression close to the expression of RHS in power symmetric basis but it does not encaptulate the expression containing the shift $$p_{k}({\bf{x+y}})$$.