How can I find the particular solution to following recurrence relation for $F_n(x,r)$?:
$$nF_n +(2n-1)xrF_{n-1}+(n-1)r^2F_{n-2}=1$$ Withe initial values $F_0=0, F_1=1$.
The corresponding homogeneous equation is the same as the recurrence relation for the Legendre Polynomials, except with the $r$ prefactors. The homogeneous solution is $(-r)^n W_{n-1}(x)$, where $$W_{n-1}=\sum_{k=1}^n\frac{P_{n-k}P_{k-1}}{k};\qquad W_{-1}=0$$
and $P_n$ are the Legendre polynomials. The full solution is a homogeneous solution plus a particular solution; I don't know if this helps. The first few orders don't seem to have an obvious pattern...
Any ideas or hints or references are much appreciated.
(This problem came across in research)