As part of my research I have to analyze recurrence relations of the form

$$f_{m,n} = af_{m-1,n} + bf_{m,n-1} + c,$$

where $a,b,c$ are any given real numbers and $f_{m,0}$ and $f_{0,n}$ any given functions (e.g. $f_{m,0} = 2^m$ and $f_{0,n} = n+1$).

Could somebody please suggest some good source (e.g. a website, a book or a paper) I could use to gain insight into this topic? Any hint will be appreciated!

Using generating functions and combinatorical arguments I found that

\begin{align} f_{m,n} &= c\sum_{i=0}^{m-1} \sum_{j=0}^{n-1} \binom{i+j}{i} a^i b^j \\ &\qquad + a^m \sum_{j=0}^{n} \binom{m+j-1}{j} b^j f_{0,n-j} \\ &\qquad + b^n \sum_{i=0}^{m} \binom{n+i-1}{i} a^i f_{m-i,0} \\ &\qquad - a^m b^n \binom{m+n}{m} f_{0,0}. \end{align}

This is nice but not really a closed-form solution.

Thank you very much!

EDIT: For the generating function $F(x,y) = \sum_{m,n \geq 1} f_{m,n} x^m y^n$ I found that

$$F(x,y) = \frac{(1 - ax) F_1(x) + (1 - by) F_2(y) + \frac{cxy}{(1-x)(1-y)} - f_{0,0}}{1-ax-by},$$

where $F_1(x) = \sum_{m \geq 0} f_{m,0} x^m$ and $F_2(y) = \sum_{n \geq 0} f_{0,n} y^n$.