A Functional Analytical Method for Constructing Asymptotic Solutions to a Discrete Painlevé Equation

In the early ‘80s, Lew and Quarles [1] developed a novel, elegant and efficient iteration method for describing the behavior of asymptotic solutions to Freud’s equations for the recurrence coefficients of orthogonal polynomials with quartic exponential weights. These Freud equations were subsequently realized to coincide with the discrete Painlevé I equation (dPI) as well as the discrete string equations of random matrix theory.

In this talk we will review the Lew-Quarles method and describe extensions of it that we have been applying [2-4] for a deeper numerical and dynamical systems analysis of dPI and the string equations.

1, J. Lew and D. Quarles, Nonnegative solutions of a nonlinear recurrence, J.Approx. Theory 38, 357–379 (1983).

2. N. Ercolani, J. Lega, and B. Tippings, Dynamics of Nonpolar Solutions to the Discrete Painlevé I Equation, SIAM J. Appl. Dyn. Sys. 21, 1322–1351 (2022).

3. N. Ercolani, J. Lega, B. Tippings, Map enumeration from a dynamical perspective, in Recent Progress in Special Functions, Contemporary Mathematics, vol. 807, Amer. Math. Soc., Providence, RI, 2024, pp. 85–110.

4. N. Ercolani, J. Lega, and B. Tippings, The Negative Freud Orbit, preprint (2025).