Meixner polynomials in several variables satisfying bispectral difference equations

Abstract

We construct a set M d whose points parametrize families of Meixner polynomials in d variables. There is a natural bispectral involution b on M d which corresponds to a symmetry between the variables and the degree indices of the polynomials. We define two sets of d commuting partial difference operators diagonalized by the polynomials. One of the sets consists of difference operators acting on the variables of the polynomials and the other one on their degree indices, thus proving their bispectrality. The two sets of partial difference operators are naturally connected via the involution b.

The multivariate Meixner polynomials as matrix elements ofSO(d, 1) representations on oscillator states

“…In this section, the recurrence relations and the difference equations satisfied by the bivariate Meixner polynomials M (β) m,n (i, k) and R (β) m,n (i, k) are derived. These relations have been obtained by Iliev in [12]. It is however interesting to see how easily these relations follow from the group-theoretical interpretation.…”

“…m,n (i, k) are derived. These relations have been obtained by Iliev in [12]. It is however interesting to see how easily these relations follow from the group-theoretical interpretation.…”

“…In his paper [9], Griffiths defined the polynomials through a generating function and gave a proof of their orthogonality with respect to a multivariate generalization of the negative binomial distribution. The same Meixner polynomials were considered by Iliev in [12]. Using generating function arguments, Iliev established the bispectrality of these polynomials, i.e.…”

“…In this section, generating functions for the multivariate polynomials M m,n (i, k) are precisely those defined in [9] and [12]. Using the results of [12], an explicit expression of the polynomials R (β) m,n (i, k) in terms of Gel'fand-Aomoto hypergeometric series is given.…”

A superintegrable discrete oscillator and two-variable Meixner polynomials

“…The polynomials M (β) n 1 ,n 2 (x 1 , x 2 ) have an explicit expression in terms of Aomoto-Gelfand hypergeometric series [14]. This expression reads…”