Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2011, Article ID 856327, 15 pages doi:10.1155/2011/856327 Research Article Monomiality Principle and Eigenfunctions of Differential Operators Isabel Cac ¸˜ ao 1 and Paolo E. Ricci 2 1 Department of Mathematics, Aveiro University, 3810-193 Aveiro, Portugal 2 Dipartimento di Matematica “Guido Castelnuovo”, Sapienza Universit` a di Roma, 00185 Roma, Italy Correspondence should be addressed to Paolo E. Ricci, riccip@uniroma1.it Received 28 December 2010; Accepted 17 March 2011 Academic Editor: B. N. Mandal Copyright q 2011 I. Cac ¸˜ ao and P. E. Ricci. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We apply the so-called monomiality principle in order to construct eigenfunctions for a wide set of ordinary differential operators, relevant to special functions and polynomials, including Bessel functions and generalized Gould-Hopper polynomials. 1. Introduction In many paper the so-called monomiality principle, introduced by Dattoli et al. 1, was used in order to study in a standard way the most important properties of special polynomials and functions 2. In this paper, we show that the abstract framework of monomiality can be used even to find in a constructive way the eigenfunctions of a wide set of linear differential operators connected with the Laguerre-type exponentials introduced in 3. In this paper, we limit ourselves to consider the first Laguerre derivative D L : DxD, so that we substitute the derivative D and multiplication operator x. with the corresponding derivative and multiplication operators  P and  M, relevant to a given set of special polynomials or functions. The same procedure could be generalized by considering for any integer n the higher-order Laguerre derivatives D nL : Dx ··· DxDxD containing n  1 ordinary derivatives, showing that this method can be used to obtain eigenfunctions for each one of the infinite many operators obtained by using the same substitutions described before. It can be noticed that this gives a further proof of the power of the monomiality technique.