论文标题
广义的六弗里达重量
A Generalised Sextic Freud Weight
论文作者
论文摘要
我们讨论了正交多项式的复发系数，相对于广义的六频\ [ω（x; t; t，λ）= | x | x |^{2λ+1} \ exp \ left（-x^6+tx^6+tx^2 \ right） $ t \ in \ mathbb {r} $。我们表明，这些复发关系中的系数可以用广义超几何函数的wronskian表示$ {} _ 1f_2（a_1; b_1; b_1，b_2; z）$。我们得出了一个非线性离散以及由复发系数满足的微分方程系统，并使用它们来研究其渐近行为。最后，我们强调了广义的四分之一，六，八分和十分弗洛伊德权重之间的引人入胜的联系，以广义超几何函数表达第一瞬间。
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.
