Consider the coefficients $C_p$ ($p=1,2,...$) defined by the recursive relation
$(p-E)\; C_p = A\left(\sqrt{p} \; C_{p-1} + \sqrt{p+1} \; C_{p+1}\right)$
where A is a fixed parameter E is an unknown parameter. I'm trying to understand
(1) if it is possible to find an explicit expression for $C_p$ and
(2) if, for some values of $E$, the sum of the series $C_p$ converges to a finite value.
(The second requirement is due to the fact that later I want to re-normalize the coefficients such that $\sum_p^\infty |C_p|^2=1$)
So far I have tried to guess the solution by requiring these two conditions to be valid at the same time
(1) $\sqrt{p} \; C_{p-1} \propto (p-E) C_p$
(2) $\sqrt{p+1} \; C_{p+1} \propto (p-E) C_p$
For each condition I can derive a recursive solution for $C_p$, but these two solutions seem to be incompatible with each other.
Does anybody have some suggestion on how to tackle this problem?