Suppose that you have two sums like $$x=\sum_{k\geq 0} \sum_{p+q=k} x^{p, q}\quad \textrm{and}\quad y=\sum_{l\geq 0} \sum_{r+s=l} y^{r, s}.$$ What would it be expression for the product $$x\cdot y=?$$
I know I'm hiding the underlying algebraic structure but just think of $\cdot$ as the usual multiplication of real numbers except for the fact that it is non-commutative.
Thanks.
$$\begin{align}x&=(x^{0,0})+(x^{1,0}+x^{0,1})+(x^{2,0}+x^{1,1}+x^{0,2})+\cdots\\&=(x^{0,0}+x^{1,0}+x^{2,0}+\cdots)+(x^{1,0}+x^{1,1}+x^{1,2}+\cdots)+\cdots\\&=\sum_p\sum_q x^{p,q}\end{align}$$And similarly for $y$. So $$x\cdot y=\sum_{p\ge0}\sum _{q\ge0}\sum_{r\ge0}\sum_{s\ge0}x^{p,q}\cdot y^{r,s}$$