In a calculation of some momenta of random matrices (GOE), I encounter a functional equation, in the form of a second-order recursion, $$L(s+1)=L(s)+2s(2s+1)L(s-1).$$ Is it familiar to someone ? Is there an explicit formula for the solutions ? By explicit, I mean in terms of the Gamma function for instance.
Just in case, I am interested in the solution generated by $L(0)=1$ and $L(1)=-\frac18$ .
As already pointed out by @KStarGamer in the comment above, Mathematica can sum the recursion using RSolve[]. The result can be rewritten in terms of the regularized incomplete Euler beta function $I_x(a,b)$ and reads, for arbitrary $L_0=L(0)$ and $L_1=L(1)$,
\begin{align}
\tag{1a}\label{eq:1a}
L(s)
&= 2^s s!\left[\frac{2 L_0-L_1}{\sqrt{2}}
\big(1 - I_{-1}\left(s+1,\tfrac{1}{2}\right)\big)
-L_0 + L_1\right] \\
\tag{1b}\label{eq:1b}
&= 2^s s!\left[\frac{2 L_0-L_1}{\sqrt{2}}
I_{2}\left(\tfrac{1}{2},s+1\right)
-L_0 + L_1\right]
\end{align}
According to Wikipedia,
The regularized incomplete beta function is the cumulative distribution function of the beta distribution, and is related to the cumulative distribution function $F(k;n,p)$ of a random variable $X$ following a binomial distribution with probability of single success $p$ and number of Bernoulli trials $n$: $$F(k;\,n,p)=\Pr \left(X\leq k\right)=I_{1-p}(n-k,k+1)=1-I_{p}(k+1,n-k).$$
This might link to the OP's original problem.
