Meixner $d$-Orthogonal Polynomials Arising from $\mathfrak{su}(1,1)$

TL;DR

This paper introduces a new family of Meixner-type d-orthogonal polynomials linked to the Lie algebra su(1,1), detailing their properties and mathematical structure through coherent states and orthogonality relations.

Contribution

It presents a novel family of Meixner-type d-orthogonal polynomials connected to su(1,1), with explicit properties and their derivation from Lie algebra representations.

Findings

Defined new Meixner d-orthogonal polynomials

Established their connection to su(1,1) algebra

Derived recurrence relations and orthogonality properties

Abstract

In this study, we present a novel family of Meixner-type $d$-orthogonal polynomials, which are distinguished as a particular subset of multiple orthogonal polynomials. We demonstrate their connection to the Lie algebra $\mathfrak{su}(1,1)$ by identifying them as matrix elements of an appropriately defined nonlinear operator. Utilizing Barut-Girardello coherent states, we explicitly outline their key features, including recurrence relations, generating functions, and $d$-orthogonality relations, among others.