Discrete orthogonal ensemble on the exponential lattices

TL;DR

This paper introduces a discrete orthogonal ensemble on exponential lattices, constructing symmetric configuration spaces and explicit correlation functions, with examples from various $q$-orthogonal polynomials.

Contribution

It develops a new framework for discrete orthogonal ensembles on exponential lattices using skew symmetric kernels and explicit correlation functions.

Findings

Explicit skew inner products and skew orthogonal polynomials derived.

Configuration space symmetry established for the ensemble.

Examples include $q$-Laguerre and $q$-Jacobi cases.

Abstract

Inspired by Aomoto's $q$-Selberg integral, the orthogonal ensemble in the exponential lattice is considered in this paper. By introducing a skew symmetric kernel, the configuration space of this ensemble is constructed to be symmetric and thus, corresponding skew inner product, skew orthogonal polynomials as well as correlation functions are explicitly formulated. Examples including Al-Salam & Carlitz, $q$-Laguerre, little $q$-Jacobi and big $q$-Jacobi cases are considered.