I want to find an inverse of an infinite tridiagonal matrix:
$ A_{m,n} = (\delta_{m,n} (n-1+\Delta) + s(\sqrt{n} \delta_{m+1,n} + \sqrt{m} \delta_{m,n+1}))_{m,n} $
where $\Delta$ and $s$ are real numbers.
I don't really have experience with this kind of questions. Therefore the most promising way to solve this, i thought, was to follow E. Kılıç (Applied Mathematics and Computation 197 (2008) 345–357) and reduce the problem to a recurrence equation.
$ P_n = (n+\Delta) P_{n-1} - n s^2 P_{n-2} \qquad, P_{-1}=1 \quad, P_{0}=\Delta $
Which solution I could not figure out.
I would be glad about an advise.