Need help with the recurrence relation.

Question

everyone. While solving a diagonalization problem I have arrived at the following recurrence relation $$f_{n}=\Big(\frac{2n}{z}-\lambda\Big)f_{n-1}-f_{n-2}$$ I know that in the case $\lambda=0$ such a recurrence relation is solved by the Bessel functions $f_{n}=J_{n}(z)$. Maybe anyone knows whether there are any special functions solving the recurrence relation for the non-zero $\lambda$? If not, then can anyone suggest any place to look at? Thanks, in advance!

Answer 1

The Bessel functions $J_n(z)$ and $K_n(z)$ satisfy the recursion $$ f_n(z) = \frac{2(n-1)}{z} f_{n-1}(z) - f_{n-2}(z)$$ Thus $J_{n+1 -\lambda z/2}(z) $ and $K_{n+1-\lambda z/2}(z)$ satisfy your recursion.