A uniform asymptotic formula for orthogonal polynomials associated with \(\exp(-x^4)\)
The paper is devoted to construct an asymptotic approximation for the orthogonal polynomials \(p_n(x)\), associated with the Freud weight \(\exp(-x^4)\), \(x\in\mathbb R\). 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 \(\rho=1\), \(-1+ \varepsilon\leq w=x\lambda^{-1}\leq M\), where \[ \zeta(w)=\begin{cases} -(\textstyle{9 \over 8}\cos^{-1}w-\textstyle {3\over 8}w(2w^2+1) \sqrt{1-\omega^2})^{2/3},\;-1<w<1\\ (\textstyle {8\over 8}w(2w^2+1) \sqrt{w^2-1}- \textstyle{9\over 8} \cosh^{-1}w)^{2/3},\;w\geq 1\end{cases} \] \[ H(\zeta)= {-1\over 2\zeta^{1/2}} \cosh^{-1}w, \text{ if }w\geq 1;\;H(\zeta)= {-1\over 2(-\zeta)^{1/2}} \cos^{-1}w, \text{ if }|w|<1, \tag{1} \] and \(Ai(x)\) is the Airy function. Moreover, when \(-1+\varepsilon\leq w\leq 1-\varepsilon\), 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 \(Ai(x)\).
scientific article; zbMATH DE number 1288056
scientific article