I have the following recurrence relation: $T(i) = \sqrt{T(i-1) \left(T(i+1) + k\right)},$ with $k \geq 0$, a fixed constant.
I know that when $k=0$, we have: $T(i) = \sqrt{T(i-1) T(i+1)},$ which solution is $T(i) = e^{ai+b}$ (geometric progression), but I do not know if there is a solution for the general case.
If there is no closed-form solution to it, as it seems to be the case, is there an easy way to compute $T(i),\;1 < i < n$ given $T(1) = A$ and $T(n) = B$ ?
More generally, is there a systematic, elegant way to (numerically) compute recurrences when the initial conditions are not "contiguous" ?