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A uniform asymptotic formula for orthogonal polynomials associated with $\exp(-x^4)$. (English) Zbl 0952.41018
The paper is devoted to construct an asymptotic approximation for the orthogonal polynomials $p_n(x)$, associated with the Freud weight $\exp(-x^4)$, $x\in\Bbb R$. The following theorem extends a result of Nevai. Theorem: Let $0<\varepsilon<1$ and $0<M<\infty$ be fixed, and let $x= (4n/3)^{1/4}w$ and $\Lambda=4n/3$. Then the asymptotic formula $$p_n(x)\exp \left(-{x^4 \over 2}\right) =\sqrt 2\Lambda^{1/24} \left({\zeta \over w^2-1} \right)^{1/4} \left\{Ai \left(\Lambda^{2/3} \zeta+{H (\zeta)\over \Lambda^{1/3}} \right)+ O(n^{-\rho}) \right\},$$ holds uniformly for $\rho=1$, $-1+ \varepsilon\le w=x\lambda^{-1}\le M$, where $$\zeta(w)=\cases -(\textstyle{9 \over 8}\cos^{-1}w-\textstyle {3\over 8}w(2w^2+1) \sqrt{1-\omega^2})^{2/3},\ -1<w<1\\ (\textstyle {8\over 8}w(2w^2+1) \sqrt{w^2-1}- \textstyle{9\over 8} \cosh^{-1}w)^{2/3},\ w\ge 1\endcases$$ $$H(\zeta)= {-1\over 2\zeta^{1/2}} \cosh^{-1}w, \text{ if }w\ge 1;\ H(\zeta)= {-1\over 2(-\zeta)^{1/2}} \cos^{-1}w, \text{ if }|w|<1, \tag 1$$ and $Ai(x)$ is the Airy function. Moreover, when $-1+\varepsilon\le w\le 1-\varepsilon$, the uniform asymptotic formula (1) holds for $\rho=7/6$. Let $b_n=(4n/3)$. The following result is proved for the positive zeros $x_{n,k}$ of the polynomial $p_n(x)$ $$x_{n,k}= b_n^{1/4}+ \widetilde a_kb_n^{-5/12} 18^{-1/3}+ b_n^{-9/12} 6^{-1}-19 \widetilde a_k^2 b_n^{-13/12} 90^{-1}2^{-2/3} 3^{1/3}+ O(n^{-17/12}),$$ where $\widetilde a_k$ is the $k$th negative zero of the Airy function $Ai(x)$.

MSC:
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
42C05General theory of orthogonal functions and polynomials
33C45Orthogonal polynomials and functions of hypergeometric type
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References:
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