Trying to solve a PDE coming from the computation of some functional of Brownian motion, I have came across the following Bessel-looking like functional recurrence:

$n^2 g_n(t) + t g_n'(t) = (t/2)(g_{n+1}(t)+g_{n-1}(t))$

where the functions $g_{-n}(t)=g_n(t)$, $n$ integer, $t$ real $\ge0$, are perfectly regular (${}'={\rm d}/{\rm d}t$), and with initial conditions $g_n(0)=\delta_{n,0}$.

Of course, were the term $n^2 g_n(t)$ absent, the solution would simply be $I_n(t)$, the modified Bessel of the second kind. In fact, it is not hard to show that even with this term, the $g_n(t)$'s have the same leading-order asymptotics when $t \to 0$ or $t \to \infty$ as the $I_n(t)$'s.

Anybody has a clue to attack this ? (I have tried Laplace-Mellin transform but was unable to solve the functional recurrence, as well as many others concerning the Taylor coefficients or the Bessel-Neumann ones...)