TITLE: Generating Function for Recurrence Relation in 2 Variable
QUESTION [5 upvotes]: I have a recurrence relation with 2 variables similar to 
$$
F(n,m) = n\cdot F(n-1,m) + (n-m)\cdot F(n-1,m-1)
$$
I want to know the steps required to get the generating Function for such recurences.
I have gone through a similar post : Solving recurrence relation in 2 variables 
. But how do I solve when there is a multiplication as in the above equations ?

REPLY [4 votes]: The strategy is to look for the generating function
$$
G(x,y)=\sum_{n,m}F(n,m)x^ny^m.
$$ 
To do so:

Multiply by $x^ny^m$ each term in the identity, getting
$$
F(n,m)x^ny^m = n\cdot F(n-1,m) x^ny^m+ (n-m)\cdot F(n-1,m-1)x^ny^m.
$$
The LHS terms sum to the generating function
$G(x,y)$.
For the first terms on the RHS, use the identity
$$
nx^ny^m=\left(x^2\cdot\frac{\partial}{\partial x}+x\right) x^{n-1}y^m.
$$
For the second terms on the RHS, use the identity
$$
(n-m)x^ny^m=xy\cdot\left(x\frac{\partial}{\partial x}-y\frac{\partial}{\partial y}\right) x^{n-1}y^{m-1}.
$$
Deduce that
$$
G(x,y)=x^2\cdot\frac{\partial G(x,y)}{\partial x}+xG(x,y)+x^2y\cdot\frac{\partial G(x,y)}{\partial x}-xy^2\frac{\partial G(x,y)}{\partial y}+H(x,y),
$$
where $H(x,y)$ is a polynomial correction due to the terms of low degrees in the sums.

Whether this PDE can be solved is another story.