A Generalised Sextic Freud Weight

Abstract

We discuss the recurrence coefficients of orthogonal polynomials with respect to a generalised sextic Freud weight \[ω(x;t,λ)=|x|^{2λ+1}\exp\left(-x^6+tx^2\right),\qquad x\in\mathbb{R},\] with parameters $\lambda >-1$ and $t\in \mathbb{R}$. We show that the coefficients in these recurrence relations can be expressed in terms of Wronskians of generalised hypergeometric functions ${}_1{F}_2(a_1;b_1,b_2;z)$. We derive a nonlinear discrete as well as a system of differential equations satisfied by the recurrence coefficients and use these to investigate their asymptotic behaviour. We conclude by highlighting a fascinating connection between generalised quartic, sextic, octic and decic Freud weights when expressing their first moments in terms of generalised hypergeometric functions.

18 pages, 3 figures

Cited by in corpus (5)

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• Differential system related to Krawtchouk polynomials: iterated regularisation and Painlevé equation