On Tur\'{a}n's inequality: new general criteria, nonnegative representations and the class of generalized Chebyshev polynomials

Abstract

Originally, Tur\'{a}n's inequality states that if $(P_n(x))_{n\in\mathbb{N}_0}$ is the sequence of Legendre polynomials, then $\Delta_n(x):=P_n^2(x)-P_{n+1}(x)P_{n-1}(x)\geq0$ for all $n\in\mathbb{N}$ and $x\in[-1,1]$. Gasper specified the parameters $\alpha,\beta>-1$ for which the Jacobi polynomials $(R_n^{(\alpha,\beta)}(x))_{n\in\mathbb{N}_0}$ satisfy Tur\'{a}n's inequality. Frequently, such results rely on the specific structure of the concrete orthogonal polynomials under consideration. Therefore, special focus has been put on general criteria (whose importance was particularly emphasized by Nevai). We provide two general criteria for Tur\'{a}n's inequality in terms of the three-term recurrence relation and also deal with sharper estimations of the Tur\'{a}n determinants $\Delta_n(x)$. They extend earlier results of Szwarc and Berg--Szwarc. Applying our criteria to the class of generalized Chebyshev polynomials $(T_n^{(\alpha,\beta)}(x))_{n\in\mathbb{N}_0}$, which are the quadratic transformations of the Jacobi polynomials, we find the companion to Gasper's above-mentioned result. At this stage, we also obtain nonnegative representations of $\Delta_n(x)$. Finally, we study $2$-sieved polynomials and discuss further examples.