Asymptotic formulas for ultraspherical polynomials $P_ n^ \lambda (x)$ and their zeros for large values of $\lambda$

Abstract:

For $\lambda > - 1/2$ we denote by $P_n^{(\lambda )}(x)$ the ultraspherical polynomial of degree $n$ and by $x_{n,k}^{(\lambda )}$ and ${h_{n,k}}(k = 1,2, \ldots ,n)$ the $k$th zeros of $P_n^{(\lambda )}(x)$ and of the Hermite polynomial ${H_n}(x)$, respectively. In this paper we establish the following formulas \[ {\lambda ^{ - n/2}}P_n^{(\lambda )}\left ( {\frac {x}{{\sqrt \lambda }}} \right ) = \sum \limits _{j = 0}^{n - 1} {{\lambda ^{ - j}}{Q_{nj}}(x) {\text {for}} \lambda \ne 0} \] and \[ x_{n,k}^{(\lambda )} = {h_{n,k}}{\lambda ^{ - 1/2}} - \frac {{{h_{n,k}}}}{8}(2n - 1 + 2h_{n,k}^2){\lambda ^{ - 3/2}} + {h_{n,k}}\left ( {\frac {{12{n^2} - 12n + 1}}{{128}} + \frac {{5n - 2}}{{24}}h_{n,k}^2 + \frac {5}{{96}}h_{n,k}^4} \right ){\lambda ^{ - 5/2}} + O({\lambda ^{ - 7/2}}),\lambda \to \infty \] where ${Q_{n0}}(x) = {H_n}(x)/n!$ and ${Q_{nj}}(x)(j = 1,2, \ldots ,n - 1)$ are polynomials specified in Theorem 1. Finally we show that the positive (negative) zeros of $P_n^{(\lambda )}(x)$ are convex (concave) functions of $\lambda$, provided $\lambda$ is sufficiently large.