Bispectrality of Meixner type polynomials

Meixner type polynomials (q(n))(n >= 0) are defined from the Meixner polynomials by using Casoratian determinants whose entries belong to two given finite sets of polynomials (S-h)(h=1)(m1)and (T-g)(g=1)(m2). They = are eigenfunctions of higher order difference operators but only for a careful choice of the polynomials (S-h)(h=1)(m1) and (T-g )(g=1)(m2), the sequence (q(n))(n >= 0) is orthogonal with respect to a measure. In this paper, we prove that the Meixner type polynomials (q(n))(n >= 0) always satisfy higher order recurrence relations (hence, they are bispectral). We also introduce and characterize the algebra of difference operators associated to these recurrence relations. Our characterization is constructive and surprisingly simple. As a consequence, we determine the unique choice of the polynomials (S-h)(h=1)(m1) and (T-g)(g=1)(m2) such that the sequence (q(n))(n >= 0) is orthogonal with respect to a measure. (C) 2020 Published by Elsevier Inc.

Automated summary

This paper discusses the Meixner type polynomials and their properties, such as being eigenfunctions of higher order difference operators and satisfying higher order recurrence relations. The characterization of the algebra of difference operators associated with these recurrence relations is constructive and surprisingly simple. Unique choices of polynomials are determined to ensure orthogonality of the sequence with respect to a measure.