Operations on formal power series in two variables.

Question

I am wondering about the operations that can be done on a formal power series in two variables. For instance, let $p\left(x,y\right)=\sum _{n=0}^{\infty}\sum _{m=0}^{\infty}c_{nm}x^ny^m$. We can obtain $$p_{\textrm{even}}\left(x,y\right)=\sum _{n+m\textrm{ is even}}c_{nm}x^ny^m$$ as $$p_{\textrm{even}}\left(x,y\right)=\frac{p\left(x,y\right)+p\left(-x,-y\right)} {2}$$ and we can make countless other operations, for instance $$p_{\textrm{translate }k,l}\left(x,y\right)=\sum _{n=k}^{\infty}\sum _{m=l}^{\infty}c_{n-k,m-l}x^ny^m$$ as $$p_{\textrm{translate }k,l}\left(x,y\right)=x^ky^lp\left(x,y\right)$$ Are there operations we can do to get $$p_{\textrm{diag}}\left(x,y\right)=\sum _{n=0}^{\infty}c_{nn}x^ny^n$$ in terms of $p\left(x,y\right)$? So far the only thing I can think of is if $c_{nm}=0$ whenever $n<m$ (we could call it "upper triangular" in analogy to a matrix) then $$p_{\textrm{diag}}\left(x,y\right)=\lim_{c\rightarrow 0}p\left(cx,c^{-1}y\right)$$ Is there a name for this kind of process? The links between power series operations and operations on their coefficients?

Answer 1

Hint: Using the coefficient of operator $[y^n]$ to denote the coefficient of $y^n$ of a series we can extract the diagonal from $p(x,y)$ via \begin{align*} G(t)=[y^0]p\left(\frac{t}{y},y\right)=\sum_{n} a_{n,n}t^n \end{align*}

This is discussed in R.P. Stanleys Enumerative Combinatorics section 6.3. Diagonals.

Related information can be found in this paper. Interesting information about diagonalisation is also given in this MO post.