Recurrence relation
I am trying to find approximate solutions $T(n)$ of the recurrence relation $$ p\ T(n-1) - \left[p+q+\overline{S} + \varepsilon \tilde{S}(n)\right]T(n) + q\ T(n+1) = 0,\\ \text{for } n=2,\dotsc,M-1 $$ with the boundary conditions $$ p\ T(1) - q\ T(2) = \alpha,\\ p\ T(M-1) - (\beta + q)\ T(M) = 0, $$ where $\varepsilon \ll 1$ is a small parameter of the system, $p,q,\alpha,\beta,\overline{S} \ge 0$ are all constants, with $p\ge q$, and the coefficients $\tilde{S_n}$ are "sparse" in the following sense: if for some positive integer $i$, $n = i\ \Delta$, then $\tilde{S}(n) > 0$; otherwise, $\tilde{S}(n) = 0$. Here $\Delta \in \mathbb{N}$ is the separation between the indices at which $\tilde{S}$ is nonzero, and we denote the number of such indices by $N$ and assume $N \ll M$.
Ideally, it is possible to find solutions as a power series in $\varepsilon$ and/or $N$ (or some combination), valid as $\varepsilon \to 0$ and $M, N \to \infty$.
What I have tried
My attempts have focussed on expressing the ratio $r_n := T(n+1)/T(n)$ as the finite continued fraction (following e.g. Risken Ch.9), $$ -q\ r_n = \cfrac{a}{ b_{n+1} + \cfrac{a}{ b_{n+2} + \cfrac{a}{ b_{n+3} + \cfrac{\ddots}{ \ddots \cfrac{a}{ b_{M-1} + \cfrac{a}{b_M} }}}}}, $$ where \begin{align} &a = -p\ q,\\ &b_k = p + q + \overline{S} + \varepsilon\tilde{S}(k), \quad \text{ for } k = n+1,\dotsc,M-1,\\ &b_M = \beta + q. \end{align} Then one can use the Euler formula, after a little bit of rearranging, to write the continued fraction as a sum of products: $$ \frac{b_{n+1}}{p} r_n = 1 + \sum_{j=n+1}^{M-1}\left(\prod_{k=n+1}^j R_k \right),$$ where \begin{align} R_{n+1} &= \frac{p\ q}{\left[p+q+\overline{S}+\varepsilon\tilde{S}(n+1)\right]\left[p+q+\overline{S}+\varepsilon\tilde{S}(n+2)\right]},\\ R_{n+2} &= \frac{p\ q}{p+q+\overline{S}+\varepsilon\tilde{S}(n+3)},\\ R_{k} &= p\ q\frac{p+q+\overline{S}+\varepsilon\tilde{S}(k-1)}{p+q+\overline{S}+\varepsilon\tilde{S}(k+1)}, \text{ for } k = n+3,\dotsc,M-1. \end{align}
This is about as far as I got. The idea is to use the sparsity (as described above) of $\tilde{S}(n)$ to simplify the solution when expressed as a sum of products, maybe making it possible to extract an approximate solution. However, it is not clear to me exactly how to proceed.
Perhaps someone can see a way forward with this, or suggest a different approach.
Thanks.