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A uniform asymptotic formula for orthogonal polynomials associated with $exp\left(-{x}^{4}\right)$. (English) Zbl 0952.41018

The paper is devoted to construct an asymptotic approximation for the orthogonal polynomials ${p}_{n}\left(x\right)$, associated with the Freud weight $exp\left(-{x}^{4}\right)$, $x\in ℝ$. The following theorem extends a result of Nevai.

Theorem: Let $0<\epsilon <1$ and $0 be fixed, and let $x={\left(4n/3\right)}^{1/4}w$ and ${\Lambda }=4n/3$. Then the asymptotic formula

${p}_{n}\left(x\right)exp\left(-\frac{{x}^{4}}{2}\right)=\sqrt{2}{{\Lambda }}^{1/24}{\left(\frac{\zeta }{{w}^{2}-1}\right)}^{1/4}\left\{Ai\left({{\Lambda }}^{2/3}\zeta +\frac{H\left(\zeta \right)}{{{\Lambda }}^{1/3}}\right)+O\left({n}^{-\rho }\right)\right\},$

holds uniformly for $\rho =1$, $-1+\epsilon \le w=x{\lambda }^{-1}\le M$, where

$\zeta \left(w\right)=\left\{\begin{array}{c}-{\left(\frac{9}{8}{cos}^{-1}w-\frac{3}{8}w\left(2{w}^{2}+1\right)\sqrt{1-{\omega }^{2}}\right)}^{2/3},\phantom{\rule{4pt}{0ex}}-1
$H\left(\zeta \right)=\frac{-1}{2{\zeta }^{1/2}}{cosh}^{-1}w,\phantom{\rule{4.pt}{0ex}}\text{if}\phantom{\rule{4.pt}{0ex}}w\ge 1;\phantom{\rule{4pt}{0ex}}H\left(\zeta \right)=\frac{-1}{2{\left(-\zeta \right)}^{1/2}}{cos}^{-1}w,\phantom{\rule{4.pt}{0ex}}\text{if}\phantom{\rule{4.pt}{0ex}}|w|<1,\phantom{\rule{2.em}{0ex}}\left(1\right)$

and $Ai\left(x\right)$ is the Airy function. Moreover, when $-1+\epsilon \le w\le 1-\epsilon$, the uniform asymptotic formula (1) holds for $\rho =7/6$. Let ${b}_{n}=\left(4n/3\right)$. The following result is proved for the positive zeros ${x}_{n,k}$ of the polynomial ${p}_{n}\left(x\right)$

${x}_{n,k}={b}_{n}^{1/4}+{\stackrel{˜}{a}}_{k}{b}_{n}^{-5/12}{18}^{-1/3}+{b}_{n}^{-9/12}{6}^{-1}-19{\stackrel{˜}{a}}_{k}^{2}{b}_{n}^{-13/12}{90}^{-1}{2}^{-2/3}{3}^{1/3}+O\left({n}^{-17/12}\right),$

where ${\stackrel{˜}{a}}_{k}$ is the $k$th negative zero of the Airy function $Ai\left(x\right)$.

##### MSC:
 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 42C05 General theory of orthogonal functions and polynomials 33C45 Orthogonal polynomials and functions of hypergeometric type