A generalization of the Mehta-Wang determinant and Askey-Wilson polynomials

TL;DR

This paper generalizes classical determinant and Pfaffian formulas related to special functions, connecting them to Askey-Wilson polynomials through $q$-analogues and moment sequences, extending their applications.

Contribution

It introduces a unified generalization of Mehta-Wang and Nishizawa formulas using little $q$-Jacobi polynomial moments, expressing results explicitly with Askey-Wilson polynomials.

Findings

Explicit evaluation of the generalized determinant in terms of Askey-Wilson polynomials.

Extension of $q$-analogues of classical formulas involving Gamma functions.

Connection of determinant formulas to special functions and orthogonal polynomials.

Abstract

Motivated by the Gaussian symplectic ensemble, Mehta and Wang evaluated the $n$ by $n$ determinant $\det((a+j-i)\Gamma(b+j+i))$ in 2000. When $a=0$, Ciucu and Krattenthaler computed the associated Pfaffian $\Pf((j-i)\Gamma(b+j+i))$ with an application to the two dimensional dimer system in 2011. Recently we have generalized the latter Pfaffian formula with a $q$-analogue by replacing the Gamma function by the moment sequence of the little $q$-Jacobi polynomials. On the other hand, Nishizawa has found a $q$-analogue of the Mehta--Wang formula. Our purpose is to generalize both the Mehta-Wang and Nishizawa formulae by using the moment sequence of the little $q$-Jacobi polynomials. It turns out that the corresponding determinant can be evaluated explicitly in terms of the Askey-Wilson polynomials.