Higher-order discrete Painlevé XXXIV for Freud ensembles with m > 3

Determine whether, for integer m > 3, the recurrence coefficients Bn of the monic orthogonal polynomials associated with the discontinuous Freud weight w(x; a) = exp(-x^{2m}) · 1_{(-a,a)^c}(x) (so that x Pn(x; a) = Pn+1(x; a) + Bn Pn-1(x; a)) satisfy, after an appropriate change of variables such as wn = 4Bn/a^2, the (2m)th-order equation in the discrete Painlevé XXXIV hierarchy of Cresswell and Joshi (1999).

Background

The paper studies symmetric gap probabilities for Freud unitary ensembles with weights w0(x) = exp(-x^{2m}) supported on the complement of a symmetric interval, leading to monic orthogonal polynomials with respect to w(x; a) = w0(x) 1_{(-a,a)^c}(x). Using the ladder-operator approach, the authors derive difference and differential-difference equations for the recurrence coefficients Bn.

For m = 1, 2, and 3, they show that, after a minor variable change (e.g., wn = 4Bn/a^2), the recurrence coefficients satisfy the discrete Painlevé XXXIV hierarchy at orders 2, 4, and 6, respectively. Motivated by these cases, they conjecture that for all higher even exponents (m > 3), the analogous recurrence coefficients obey the (2m)th-order member of the same hierarchy.