A formula for the solution of some kind of homogeneous recurrence of variable coefficients with two indices (Q2719615)

The author considers the following linear recurrence of variable coefficients with two indices NEWLINE\[NEWLINE\begin{cases} u_{i,j} =f (i, j) u_{i-1,j-1} +g(i,j)u_{i-q, j-q}\\ u_{i,0}=c_{i,0},u_{0,j} =c_{0,j} (i,j =0, 1, \dots),\;u_{i,j}= 0\;(i<0\text{ or }j<0)\end{cases}NEWLINE\]NEWLINE where \(i,j=1,2, \dots,q \geq 2\), \(f(i,j)\) and \(g(i,j)\) are variable numbers, \(c_{i,0}\) and \(c_{0,j}\) are arbitrary constants. He gives its general solution as NEWLINE\[NEWLINEu_{i,j}= \begin{cases} F(i,j; j,j -qm)c_{i-j,0} & (i>j)\\ F(i,j;i,i-qm)c_{0,j-i} & (j\geq i) \end{cases}NEWLINE\]NEWLINE where \(F\) is a complex function.