Asymptotic formulas for ultraspherical polynomials 𝑃_{𝑛}^{𝜆}(𝑥) and their zeros for large values of 𝜆

Abstract

For λ > − 1 / 2 \lambda > - 1/2 we denote by P n ( λ ) ( x ) P_n^{(\lambda )}(x) the ultraspherical polynomial of degree n n and by x n , k ( λ ) x_{n,k}^{(\lambda )} and h n , k ( k = 1 , 2 , … , n ) {h_{n,k}}(k = 1,2, \ldots ,n) the k k th zeros of P n ( λ ) ( x ) P_n^{(\lambda )}(x) and of the Hermite polynomial H n ( x ) {H_n}(x) , respectively. In this paper we establish the following formulas \[ λ − n / 2 P n ( λ ) ( x λ ) = ∑ j = 0 n − 1 λ − j Q n j ( x ) for λ ≠ 0 {\lambda ^{ - n/2}}P_n^{(\lambda )}\left ( {\frac {x}{{\sqrt \lambda }}} \right ) = \sum \limits _{j = 0}^{n - 1} {{\lambda ^{ - j}}{Q_{nj}}(x)\,{\text {for}}\,\lambda \ne 0} \] and \[ x n , k ( λ ) = h n , k λ − 1 / 2 − h n , k 8 ( 2 n − 1 + 2 h n , k 2 ) λ − 3 / 2 + h n , k ( 12 n 2 − 12 n + 1 128 + 5 n − 2 24 h n , k 2 + 5 96 h n , k 4 ) λ − 5 / 2 + O ( λ − 7 / 2 ) , λ → ∞ x_{n,k}^{(\lambda )} = {h_{n,k}}{\lambda ^{ - 1/2}} - \frac {{{h_{n,k}}}}{8}(2n - 1 + 2h_{n,k}^2){\lambda ^{ - 3/2}} + {h_{n,k}}\left ( {\frac {{12{n^2} - 12n + 1}}{{128}} + \frac {{5n - 2}}{{24}}h_{n,k}^2 + \frac {5}{{96}}h_{n,k}^4} \right ){\lambda ^{ - 5/2}} + O({\lambda ^{ - 7/2}}),\lambda \to \infty \] where Q n 0 ( x ) = H n ( x ) / n ! {Q_{n0}}(x) = {H_n}(x)/n! and Q n j ( x ) ( j = 1 , 2 , … , n − 1 ) {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 ( λ ) ( x ) P_n^{(\lambda )}(x) are convex (concave) functions of λ \lambda , provided λ \lambda is sufficiently large.