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 $$p_{n+1}(x)=\frac{1}{x}I^2(x\cdot p_n(x))+I(\frac{1}{x^2}I^3((1+\frac{x}{4})\cdot p_{n-1}(x))),$$ 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: $\mathbb{E}(X^n_t)=p_n(t)$