My question is about the topological algebra $F$ of formal power series with coefficients in $\mathbb{C}$, which is the set of formal power series with multiplication given by the Cauchy product.
The index of a formal power series $\sum_{n=0}^\infty a_n x^n$ is the smallest natural number $t$ such that $a_t \not= 0$. The topology on $F$ is given by the metric $d(A, B) = 2^{-k}$, where $k$ is the index of $A-B$.
It is my understanding that any sequence $\{L_n\}_{n \in \mathbb{N}}$ of formal power series such that the index of $L_n$ equals $n$ forms a pseudobasis of $F$ in the sense that if $A \in F$ then there exists a unique sequence of $a_n \in \mathbb{C}$ such that $\sum_{n=0}^\infty a_n L_n$ converges to A.
Now let $L_n = \sum_{k=n}^\infty l_{n,k} x^k$, where $x$ is the formal variable, so that $l_{n,k}$ is an infinite triangular array. My question is whether there is a formula in terms of $l_{n,k}$ for the array $\theta_{i,j}$ defined by $ \sum_{j=0}^\infty \sum_{i=j}^\infty \theta_{i,j} L_i = (\sum_{n=0}^\infty \sum_{k=n}^\infty l_{n,k} x^k)(\sum_{\nu=0}^\infty \sum_{r=\nu}^\infty l_{\nu,r} x^r).$
I have tried just working it out order by order in $x$ but I can't seem to make sense of it.