I have a nasty linear recurrence equation for a sequence $u_{n,m}$ for natural numbers $n,m$.
In the equation I have the term
$$\sum_{i=0}^{m-1} u_{n,i}C_i$$
Where $C_i$ are the Catalan numbers. My recurrence equation may not have closed form, but I would be happy to obtain a generating function for $u$ with fixed $n$. The Catalan generating function is well known.
What procedure would help me find a functional equation? I mean, is there a transform (Laplace, Fourier or others) that express such product into a basic operation in the functional space?
For the sake of completeness, I am looking the number of paths inside a precise convex polygon, from one precise point to another.