$q$-Discrete Painlevé equations for recurrence coefficients of modified $q$-Freud orthogonal polynomials

Abstract

We present an asymmetric $q$-Painlevé equation. We will derive this using $q$-orthogonal polynomials with respect to generalized Freud weights: their recurrence coefficients will obey this $q$-Painlevé 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 $\alpha -q-P_V$.

16 pages, 4 figures. Accepted, to appear in Journal of Difference Equations and Applications