Symmetric Sextic Freud Weight

This paper investigates properties of the sequence of coefficients $(\beta_n)_{n\geq0}$ in the recurrence relation satisfied by the sequence of monic symmetric polynomials, orthogonal with respect to the symmetric sextic Freud weight $$\omega(x;\tau, t) = \exp(-x^6 +\tau x^4 + t x^2), \qquad x \in \mathbb{R},$$ with real parameters $\tau$ and $t$. It is known that the recurrence coefficients $\beta_n$ satisfy a fourth-order nonlinear discrete equation, which is a special case of the second member of the discrete Painlev\'{e} I hierarchy, often known as the ''string equation''. The recurrence coefficients have been studied in the context of Hermitian one-matrix models and random symmetric matrix ensembles with researchers in the 1990s observing ''chaotic, pseudo-oscillatory'' behaviour. More recently, this ''chaotic phase'' was described as a dispersive shockwave in a hydrodynamic chain. Our emphasis is a comprehensive study of the behaviour of the recurrence coefficients as the parameters $\tau$ and $t$ vary. Extensive computational analysis is carried out, using Maple, for critical parameter ranges, and graphical plots are presented to illustrate the behaviour of the recurrence coefficients as well as the complexity of the associated Volterra lattice hierarchy. The corresponding symmetric sextic Freud polynomials are shown to satisfy a second-order differential equation with rational coefficients. The moments of the weight are examined in detail, including their integral representations, differential equations, and recursive structure. Closed-form expressions for moments are obtained in several special cases in terms of generalised hypergeometric functions and modified Bessel functions. The results highlight the rich algebraic and analytic structures underlying the Freud weight and its connections to integrable systems.