$q$-Discrete Painlev\'e equations for recurrence coefficients of modified $q$-Freud orthogonal polynomials

TL;DR

This paper derives an asymmetric q-Painlevé equation from q-orthogonal polynomials related to generalized Freud weights, providing a stable computational method for recurrence coefficients and linking to the alpha-q-PV equation.

Contribution

It introduces a new asymmetric q-Painlevé equation derived from q-orthogonal polynomials and connects it to existing Painlevé equations, with a stable method for computing solutions.

Findings

Derived an asymmetric q-Painlevé equation from q-orthogonal polynomials.

Established a stable computational method for recurrence coefficients.

Connected the new equation to alpha-q-PV.

Abstract

We present an asymmetric $q$-Painlev\'e equation. We will derive this using $q$-orthogonal polynomials with respect to generalized Freud weights: their recurrence coefficients will obey this $q$-Painlev\'e equation (up to a simple transformation). We will show a stable method of computing a special solution which gives the recurrence coefficients. We establish a connection with $\alpha-q-P_V$.