Smallest zeros of some types of orthogonal polynomials: asymptotics
Let the measure \(d\mu_0(x)=e^{-x^2}dx\) on \textbf{R} have orthogonal polynomials \(H_n(x)=2^nx^n+\cdots\) (the classical Hermite polynomials) and the measure \(d\mu_1(x)={e^{-x^2}dx \over (x^2+a^2)}\;(a\in\mathbf{R}\setminus \{0\})\) on \textbf{R} the orthogonal polynomials \(T(x)=2^nx^n+\cdots\). Finally, the orthogonal polynomials with respect to the Sobolev inner product \[ (f,g)_S:=\int_{\mathbf{R}}\, f(x)g(x)d\mu_0(x)+\lambda\, \int_{\mathbf{R}}\,f'(x)g'(x) d\mu_1(x)\;(\lambda>0,\,a\in\mathbf{R}) \] are denoted by \(Q_n(x)=2^nx^n+\cdots\). The author then proves Mehler-Heine type formulas \[ \lim_{n\rightarrow\infty}\, {(-1)^nT_{2n}(x/(2\sqrt{n})) \over 2^{2n}(n-1)!}={| a| \over \sqrt{\pi}}\, \cos{x},\;\lim_{n\rightarrow\infty}\, {(-1)^nT_{2n+1}(x/(2\sqrt{n})) \over 2^{2n+1}n!}={| a| \over \sqrt{\pi}}\, \sin{x}, \] uniformly on compact subsets of \textbf{C}, and \[ \lim_{n\rightarrow\infty}\, {(-1)^nQ_{2n}(x/(2\sqrt{n})) \over 2^{2n}(n-1)!}={\varphi(1+2\lambda)\over \varphi(1+2\lambda)-1}\,{| a| \over \sqrt{\pi}}\, \cos{x}, \] and \[ \lim_{n\rightarrow\infty}\, {(-1)^nQ_{2n+1}(x/(2\sqrt{n})) \over 2^{2n+1}n!}={\varphi(1+2\lambda)\over \varphi(1+2\lambda)-1}\,{| a| \over \sqrt{\pi}}\, \cos{x}, \] both uniformly on compact subsets of \textbf{C}, where \(\varphi(x)=x+\sqrt{x^2-1}\). Moreover, a result on relative asymptotics is established in the form \[ \lim_{n\rightarrow\infty}\, \sqrt{\left[{n\over 2}\right]}\,{Q_n(x)\over H_n(x)}\,=\, \begin{cases} {\varphi(1+2\lambda)\over \varphi(1+2\lambda)-1}\,(-ix+| a| ),&\text{if }x\in\mathbf{C}_{+},\cr {\varphi(1+2\lambda)\over \varphi(1+2\lambda)-1}\,(ix+| a| ),&\text{if }x\in\text\textbf{C}_{-},\end{cases} \] uniformly on compact subsets of the half-planes \textbf{C}\(_{+}\),\textbf{C}\(_{-}\), respectively. Finally, several results on the smallest zeros of \(T_n\) and \(Q_n\) are derived. A nicely written paper with interesting results.
scientific article; zbMATH DE number 2173200