A set of finite order differential equations for the Appell polynomials

Let {Rn(x)}n=0∞ denote the set of Appell polynomials which includes, among others, Hermite, Bernoulli, Euler and Genocchi polynomials. In this paper, by introducing the generalized factorization method, for each k∈Nk∈N, we determine the differential operator {Ln,k(x)}n=0∞ such that Ln,k(x)(Rn(x))=λn,kRn(x), where λn,k=(n+k)!n!−k!. The special case k=1k=1 reduces to the result obtained in [M.X. He, P.E. Ricci, Differential equation of Appell polynomials via the factorization method, J. Comput. Appl. Math. 139 (2002) 231–237]. The differential equations for the Hermite and Bernoulli polynomials are exhibited for the case k=2k=2.