Bispectrality of Meixner type polynomials

Abstract

Meixner type polynomials (qn)n≥0 are defined from the Meixner polynomials by using Casoratian determinants whose entries belong to two given finite sets of polynomials (Sh)h=1m1 and (Tg)g=1m2. They are eigenfunctions of higher order difference operators but only for a careful choice of the polynomials (Sh)h=1m1 and (Tg)g=1m2, the sequence (qn)n≥0 is orthogonal with respect to a measure. In this paper, we prove that the Meixner type polynomials (qn)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 (Sh)h=1m1 and (Tg)g=1m2 such that the sequence (qn)n≥0 is orthogonal with respect to a measure.