Orthogonal polynomials, q-difference equations and discrete Painle´ve.

stract: I discuss recent work [1, 2] on finding an accurate asymptotic approximation of orthogonal polynomials whose measure is supported on the discrete q-lattice. In searching for these asymptotic results, deep connections between q-difference calculus, the Riemann-Hilbert problem and discrete Painlevé equations are unearthed. We will touch on some of these connections and how they show themselves in the context of orthogonal polynomials. References: [1] N. Joshi, and T. Lasic Latimer, Asymptotics of discrete q-Freud II orthogonal polynomials from the q-Riemann Hilbert problem. Nonlinearity. 36 (8): 3969, 2023. [2] T. Lasic Latimer, Asymptotics for multiple q-orthogonal polynomials from the Riemann-Hilbert Problem. arXiv:2502.00335. 2025.