A Generalised Sextic Freud Weight

英文摘要：

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 $λ>-1$ 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 ${}_1F_2(a_1;b_1,b_2;z)$. 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.