On Tur\'an 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 Tur\'{a}n 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 Tur\'{a}n inequalities for some symmetric orthogonal polynomials with a finite or infinite support defined by a three term recurrence.