Exceptional Charlier and Hermite orthogonal polynomials

2013/09/04 by Duran, Antonio J.
#33C45 #33E30 #42C05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1309.1175

Abstract

Using Casorati determinants of Charlier polynomials, we construct for each finite set F of positive integers a sequence of polynomials r_n^F, n∈ σ_F, which are eigenfunction of a second order difference operator, where σ_F is an infinite set of nonnegative integers, σ_F \varsubsetneq \NN. For certain finite sets F (we call them admissible sets), we prove that the polynomials r_n^F, n∈ σ_F, are actually exceptional Charlier polynomials; that is, in addition, they are orthogonal and complete with respect to a positive measure. By passing to the limit, we transform the Casorati determinant of Charlier polynomials into a Wronskian determinant of Hermite polynomials. For admissible sets, these Wronskian determinants turn out to be exceptional Hermite polynomials.

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