Abstract
Gomez-Ullate, Kamran and Milson have found polynomial eigenfunctions of a Sturm-Liouville problem. These polynomials, denoted by X
1
-Laguerre and X
1
-Jacobi and starting with degree one, are eigenfunctions of a second-order linear differential operator. In this paper, we investigate the X
1
-Bessel case which we denote by
. We wrote these polynomials as explicit functions of n, decompose it for the basis (x−b)
2
x
i
, and we expand
in terms of Bessel orthogonal polynomials
, using generalized Carlitz formula. Finally, we give a non-hermitian orthogonality satisfied by these polynomials.