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[Community Question] Calculus: How to solve a two-dimensional recurrence problem involving two unknown sequences?

One of our user asked:

In a mathematical physical problem, one has to deal with a non-trivial two-dimensional recurrence problem involving the two sequences $D_{i,j}$ and $\psi_{i,j}$, where $i,j = 1, 2, \dots, N$. Specifically, \begin{align} 2a \left( \psi_{i+1,i+1}+\psi_{i+1,i-1}-\psi_{i-1,i+1}-\psi_{i-1,i-1} \right) + 16 \left( \psi_{i+1,i} - \psi_{i-1,i} \right) +3a \left( D_{i+1,i+1}+D_{i-1,i+1}+D_{i-1,i-1}+D_{i+1,i-1} \right) -16 \left( D_{i,i+1}+D_{i,i-1} \right) +64 \left( D_{i-1,i} + D_{i+1,i} \right) = 0 \, , \\ -----------------------------------\\ 4(4+a)\psi_{i,i} + 2a \left( \psi_{i+1,i+1}+\psi_{i-1,i+1}+\psi_{i-1,i-1}+\psi_{i+1,i-1} \right) +8 \left( \psi_{i,i+1}+\psi_{i-1,i}+\psi_{i,i-1}+\psi_{i+1,i} \right) +3a \left( D_{i+1,i+1}+D_{i+1,i-1}-D_{i-1,i+1}-D_{i-1,i-1} \right) +24 \left( D_{i+1,i} - D_{i-1,i} \right) = 0 \, . \end{align}

Here, $a > 0$.

For a single unknown sequence the generating function method seems to be often a suitable approach for the recurrence problems. I was wondering whether this can also be applied for the present recurrence at hand to yield expressions for the two unknown sequences.

Any help or hint are very welcome.

Thank you


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