Sequences of polynomials with a 3-term recurrence relations are well known for orthogonal polynomials. Do recurrence relations using differential or integral operators also appear in some theories?
I have encountered a sequence of polynomials $p_n$ of degree $2n$ with $$p_{n+1}(x)=\frac{1}{x}I^2\left(x\cdot p_n(x)\right)+I\left(\frac{1}{x^2}I^3\left(\left(1+\frac{x}{4}\right)\cdot p_{n-1}(x)\right)\right),$$ where $I(x^n)=\frac{1}{n+1}x^{n+1}$ is the integral operator.
Have any families of polynomials with similar recurrences been studied? Note, that the upper formula can be rewritten using a differential operator.
Background
In my case the polynomials appear as moments of a stochastic process: $\mathbb{E}(X^{2n}_t)=p_n(t)$
I has started with a (moment) generating function $$exp(xz){}_1F_1(1-z,2,xz)=\sum_{n=0}^\infty p_n(x)z^{2n}$$ This leads to the explicit formula $$p_n(x)=\sum_{k=n}^{2n}\frac{\theta(3n-k+1,k-n+1)}{(3n-k)!(3n-k+1)!}x^n,$$ where $\theta(n,m)$ are Areatangenshyperbolicus numbers $$\dfrac{1}{m!}\left(2Artanh\left(\dfrac{z}{2}\right)\right)^m = \sum_{n=0}^\infty\theta(n,m)\frac{z^n}{n!}.$$ A recurrence relation for the numbers has led me to the relation for the polynomials.
