I have a product looking like this:
$$\omega = (aP+bQ+ \cdots)(cP+dQ+ \cdots)(eP+fQ+ \cdots)(gP+hQ+ \cdots)$$
So terms like $P,Q,R$, and so on repeat in every parentheses here, with different variable coefficients each time. I hope that makes sense.
Is there another way to write an expression like this, possibly in terms of $P,Q$? Is it a well known type of equation?
Edit - maybe clearer if I write general form:
$$\prod_{i=1}^{n} \sum_{j=1}^{m} x_{ij} y_j$$
$n$ is the number of products (the example I have above is $n=4$) and $m$ is the number of terms in each product.
The only thing I can think of (in this generality) is the following :
$$\prod_{i=1}^n\sum_{j=1}^mx_{i,j}y_j=\sum_{(\alpha_1,...,\alpha_m)\mid \alpha_1+...+\alpha_m=n}X_{\overline{\alpha}}y_1^{\alpha_1}...y_m^{\alpha_m} $$
Where $\overline{\alpha}$ stands for $(\alpha_1,...,\alpha_n)$ and :
$$X_{\overline{\alpha}}:=\sum_{(A_1,...,A_n)\text{ partitions of }\{1,...,m\}\text{ with } |A_i|=\alpha_i\text{ for all } i}\prod_{j=1}^mw_j(A_1,...,A_n) $$
Where $w_j(A_1,...,A_n)=x_{i,j}$ where $i$ is the unique index such that $j\in A_i$.
Clearly the formula is nasty, however it may be interesting in some cases (maybe)...