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 $F\_2(y) = \sum\_{n \geq 0} f\_{0,n} y^n$.



Linear two-dimensional recurrence relation
I guess you might not be interested in this but here is a generating function.
If $G(x,y)=\sum\_{m,n\geq0}f\_{m,n}\,x^ny^m$ then
$$G(x,y)=\frac{\frac{x^2-(b+2)x}{(1-x)^2}+\frac{1-ay}{1-2y}+\frac{cxy}{(1-x)(1-y)}}{1-ay-bx}.$$


