Non linear difference equations arising from a deformation of the q-Laguerre weight

TL;DR

This paper investigates a one-parameter deformation of the q-Laguerre weight, deriving nonlinear difference equations for orthogonal polynomial recurrence coefficients related to discrete Painleve systems.

Contribution

It introduces a new deformation of the q-Laguerre weight and derives associated nonlinear difference equations for the recurrence coefficients.

Findings

Recurrence coefficients expressed via auxiliary quantities

Derived nonlinear difference equations similar to discrete Painleve systems

Established q-analog of a sum rule for the weight

Abstract

We study, in this paper, a one parameter deformation of the $q-$Laguerre weight function. An investigation is made on the polynomials orthogonal with respect to such a weight. With the aid of the two compatibility conditions previously obtained in \cite{Chen-Ism} and the $q-$analog of a sum rule obtained in this paper, we derive expressions for the recurrence coefficients in terms of certain auxiliary quantities, and show that these quantities satisfy a pair of first order non linear difference equations. These difference equations are similar in form to the recognized asymmetric discrete Painleve systems such as $\alpha q-$P-IV and $\alpha q-$P-V.