Let $r>4$ be a positive integer. Let us consider this difference equation: $$u_{q+1}=(r^{2q+1}+(c/a))u_{q}-(c/a)r^{2q-1}u_{q-1} +2c+d-(bc/a)$$
where $a,b,c,d$ are integers.
I want to find a closed form, but I am not able to find the good idea. Or at least how we can prove that a solution exists.
Same technique as before. Expand using a symbolic algebra system on the computer. Find a pattern. Prove by induction.
$$ \begin{array} ( &\displaystyle u_n = u_1 r^{f(n)} + \sum_{k=1}^{n-1} \left[ \alpha^k (u_1 - u_0 r) + \tfrac{\alpha^k-1}{\alpha-1}\gamma \right] r^{f(n)-g(k)} \\ & \\ &f(n) = n^2-1 \\ &g(k) = k\,(k+2) \\ & \\ & \alpha = c/a \\ & \gamma = 2c + d -bc/a \\ \end{array} $$
which is fairly nice if $u_0=u_1=0$.