On Turán inequality for ultraspherical polynomials

Abstract

We show that the normalised ultraspherical polynomials, $G_n^{(\lambda )}(x)=C_n^{(\lambda )}(x)/C_n^{(\lambda )}(1)$, satisfy the following stronger version of Turán inequality, $$|x{|}^{\theta }{\left(G_n^{(\lambda )}(x)\right)}^2-G_{n-1}^{(\lambda )}(x)G_{n+1}^{(\lambda )}(x)\ge 0,|x|\le 1,$$ where $\theta =4/(2-\lambda )$ if $-1/2<\lambda \le 0$, and $\theta =2/(1+2\lambda )$ if $\lambda \ge 0$. We also provide a similar generalisation of Turán inequalities for some symmetric orthogonal polynomials with a finite or infinite support defined by a three term recurrence.

11 pages, 3 figures. The proof of Case 1 of Lemma 4 was corrected; results are unchanged