Diagonal power series is holonomic

Question

Let $F(x,y):= \sum_{n,m} a_{n,m}x^m y^{n}$. Where $m,n$ are postive integers. My question to start with is that is there any basic operation that is integration, differential or substituting $x,y$ with function of $g(x),g(y)$ such that we extract $DF(x,y):=\sum_{n} a_{n,n}x^n y^n$ ?

I feel the following is true but cannot prove. If $F(x,y)$ is a solution to a linear differential equation with over the field $\mathbb{Q}(x,y)$ then so does $DF(x,y)$.

The usual closure properties for solution to differetial equation hold true such as sum, products etc. So if I can extract the $DF(x,y)$ from $F(x,y)$ with those operation then I could prove it.

Answer 1

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

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