Differential equation of Appell polynomials via the factorization method (Q1349150)

Let \({P_n(x)}_{n=0}^\infty \) be a sequence of polynomials of degree \(n\). The authors define two sequences of differential operators \(\Phi _n \) and \(\Psi _n\) satisfying the properties \(\Phi _n(P_n(x))= P_{n-1}(x)\) and \(\Psi _n(P_n(x))= P_{n+1}(x)\), that is two operators playing a role analogous to that of derivative and multiplicative operators, respectively, on monomials. The iterations of \(\Phi _n\) and \(\Psi _n\) to \(P_n(x)\) give the relationships \[ \begin{cases} (\Phi _{n-1}\Psi _n)P_n(x)=P_n(x),\\(\Psi _{n-1}\Phi _n)P_n(x)=P_n(x),\end{cases} \quad \text{and}\quad \begin{cases} (\Phi _1 \Phi _2 \dots \Phi _{n-1}\Phi _n)P_n(x)=P_0(x),\\(\Psi _{n-1}\Psi _{n-2}\dots \Psi _1 \Psi _0)P_0(x)=P_n(x),\end{cases} \] that can be used to derive a higher order differential equation satisfied by some special polynomials. In this paper, the authors construct the operators \(\Phi _n\) and \(\Psi _n\) for Appell polynomials \(R_n(x)\) and derive the corresponding differential equation. As special cases, the differential equations for Bernoulli and Euler polynomials are given.