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

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 $\alpha-q-P_V$.