Analytical description of the transition region for recurrence coefficients

Characterize analytically the structure and size of the transition region in the sequence of recurrence coefficients βn of monic orthogonal polynomials with respect to the symmetric sextic Freud weight ω(x; τ, t) = exp(−x^6 + τ x^4 + t x^2), including its dependence on τ and κ = −t/τ^2, and ascertain the onset in n and scaling regimes where βn departs from the cubic asymptotic curve 60 β(n)^3 − 12 τ^2 − 2 t = n.

Background

The paper numerically investigates the recurrence coefficients βn associated with orthogonal polynomials for the symmetric sextic Freud weight ω(x; τ, t) = exp(−x^6 + τ x^4 + t x^2). For large n, βn follows a cubic asymptotic curve derived from Lubinsky–Mhaskar–Saff theory and specified here as 60 β(n)^3 − 12 τ^2 − 2 t = n. However, for small-to-moderate n, the authors find a parameter-dependent transition region where βn does not follow the cubic and exhibits quasi-periodic structures.

Historical numerical studies described this regime as “chaotic,” but subsequent analyses showed it is non-chaotic and quasi-periodic. The present work provides extensive numerical evidence that the transition region’s size and internal structure depend on τ and on the reduced parameter κ = −t/τ^2, and explicitly leaves the analytic characterization open.