The algebra of recurrence relations for exceptional Laguerre and Jacobi polynomials

Abstract

Exceptional Laguerre and Jacobi polynomials p n ( x ) p_n(x) are bispectral, in the sense that as functions of the continuous variable x x , they are eigenfunctions of a second order differential operator and as functions of the discrete variable n n , they are eigenfunctions of a higher order difference operator (the one defined by any of the recurrence relations they satisfy). In this paper, under mild conditions on the sets of parameters, we characterize the algebra of difference operators associated to the higher order recurrence relations satisfied by the exceptional Laguerre and Jacobi polynomials.