In $2012$, Rajkumar presented an interesting simplification of Apéry's theorem, i.e., that $\zeta(3)$ is irrational, where $\zeta$ denotes the Riemann zeta function. I am trying to understand his proof, but I severely lack knowledge on solving 2-dimensional recurrence relations in multiple homogeneous polynomials. In particular, he began by defining $$ f(i,j)=i^3+2i^2j+2ij^2+j^3, \\ g(i,j)=i^3-2i^2j+2ij^2-j^3. $$ He then constructed the recurrence relation $$ \left(\begin{array}{cc} f(i,j) & g(0,j) \\ f(0,j) & g(i,j) \end{array}\right) \left(\begin{array}{c} u_{i-1,j}\\ u_{i-1,j-1} \end{array}\right) = f(i,0) \left(\begin{array}{c} u_{i,j}\\ u_{i,j-1} \end{array}\right) $$ and showed that for integers $i, j \geq 1$ it has a rational valued solution $u_{i,j}$ for certain boundary conditions.
I would really like to understand how one solves such recurrence relations, but unfortunately after searching extensively on the internet for related articles or tutorials I have found nothing. I understand how basic recurrence relations work, but in this case I'm not sure what I'm supposed to do and what theory I can apply.
Can anyone give me any pointers?