论文标题
广义的高阶弗洛伊德重量
Generalised higher-order Freud weights
论文作者
论文摘要
我们讨论多项式正交相对于半经典的普通级弗洛伊德重量\ [ω（x; t; t，λ）= | x | x |^{2λ+1} \ exp \ left（tx^2-x^{2m} {2m} \ right），\ qquad $ t \ in \ mathbb {r} $和$ m = 2,3，\ dots $ \。 $ m = 2,3，\点$的广义高阶弗洛伊德权重的顺序形成了权重的层次结构，第一刻，相关的层次结构和复发系数。我们证明，第一个时刻可以写成通用超几何$ _1f_m $函数的有限分区和，并表明复发系数满足差异方程，这是第一个离散的Painlevé层次结构的成员。我们分析了复发系数的渐近行为，并将零为$ n \ to \ infty $的限制分布。我们还研究了由多项式和相关特性满足的结构和其他混合复发关系。
We discuss polynomials orthogonal with respect to a semi-classical generalised higher order Freud weight \[ω(x;t,λ)=|x|^{2λ+1}\exp\left(tx^2-x^{2m}\right),\qquad x\in\mathbb{R},\] with parameters $λ> -1$, $t\in\mathbb{R}$ and $m=2,3,\dots$\ . The sequence of generalised higher order Freud weights for $m=2,3,\dots$, 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 generalised hypergeometric $_1F_m$ 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 $n \to \infty$. We also investigate structure and other mixed recurrence relations satisfied by the polynomials and related properties.
