Differential equation of Appell polynomials via the factorization method

Abstract

Let { P n ( x)} ∞ n=0 be a sequence of polynomials of degree n. We define two sequences of differential operators Φ n and Ψ n satisfying the following properties: Φ n(P n(x))=P n−1(x), Ψ n(P n(x))=P n+1(x). By constructing these two operators for Appell polynomials, we determine their differential equations via the factorization method introduced by Infeld and Hull (Rev. Mod. Phys. 23 (1951) 21). The differential equations for both Bernoulli and Euler polynomials are given as special cases of the Appell polynomials.