A set of finite order differential equations for the Appell polynomials

Abstract

Let { (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 kâ̂̂N, we determine the differential operator {Ln,k(x)}n=0 such that Ln,k(x)( (x))=λn, (x), where λn, =(n+k)!n!-k!. The special case k=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=2. © 2013 Elsevier B.V. All rights reserved.