Asymptotic expansions of recursion coefficients of orthogonal polynomials with truncated exponential weights

摘要

Let $eta>0$ and $W_eta (x) = exp (-|x|^eta )$, $x in mathbb{R}$. For$c>0$, define $W_{eta, cn} (x) = W_eta (x)$ if $|x| leqc^{1/eta},a_{2n}$ and $W_{eta, cn} (x) = 0$ if $|x| >c^{1/eta},a_{2n}$, where $a_{2n}$ denotes Mhaskar-Rahmanov-Saff number for$W_eta$. Let $gamma_n (W_{eta, cn})$ be the leading coefficient of the$n$th orthonormal polynomial corresponding to $W_{eta, cn}$ and write$lpha_n(W_{eta, cn}) = gamma_{n-1}(W_{eta, cn})/gamma_{n}(W_{eta,cn})$. It is shown that if $c>1$ and $eta$ is a positive even integer then$lpha_n(W_{eta, cn})^{1/eta}$ has an asymptotic expansion. Also when$0