Generalized higher-order Freud weights

Abstract

We discuss polynomials orthogonal with respect to a semi-classical generalized higher-order Freud weight ω ( x ; t , λ ) = | x | 2 λ + 1 exp ⁡ ( t x 2 − x 2 m ) , x ∈ R , with parameters λ > − 1 , t ∈ R and m = 2 , 3 , … . The sequence of generalized higher-order Freud weights for m = 2 , 3 , … , forms a hierarchy of weights, with associated hierarchies for the first moment and the recurrence coefficient. We prove that the first moment can be written as a finite partition sum of generalized hypergeometric 1 F m functions and show that the recurrence coefficients satisfy difference equations which are members of the first discrete Painlevé hierarchy. We analyse the asymptotic behaviour of the recurrence coefficients and the limiting distribution of the zeros as n → ∞ . We also investigate structure and other mixed recurrence relations satisfied by the polynomials and related properties.

A family of integrable maps associated with the Volterra lattice

“…It is also worth pointing out that part of the original motivation for the work in [22] was to consider autonomous versions of the higher order discrete Painlevé equations from [12], and new applications of the latter have been found very recently. Non-autonomous analogues of the Volterra maps V g have been considered in the context of Hermitian matrix models [4], where they arise as string equations, and they also appear as recursion relations for orthogonal polynomials associated with generalised Freud weights of higher order [11]. In these applications, the algebro-geometric structure of the Volterra maps should be relevant to the asymptotic description of the oscillatory behaviour that is observed in specific parameter regimes.…”