Tur\'an inequalities from Chebyshev to Laguerre polynomials

TL;DR

This paper introduces a unified recursive scheme for classical polynomials, demonstrating orthogonality and Turán inequalities for specific parameter choices, thereby extending understanding of polynomial inequalities and orthogonality.

Contribution

It generalizes recursive polynomial definitions to include classical families and proves orthogonality and Turán inequalities for these polynomials under certain conditions.

Findings

Orthogonality for polynomial sequences with g(n)=n and fixed h

Turán inequalities hold for x ≥ 0 when h(n)=n^s with 0 ≤ s ≤ 1

Unified framework encompasses Chebyshev, Laguerre, and Nekrasov--Okounkov polynomials

Abstract

Let $g$ and $h$ be real-valued arithmetic functions, positive and normalized. Specific choices within the following general scheme of recursively defined polynomials \begin{equation*} P_n^{g,h}(x):= \frac{x}{h(n)} \sum_{k=1}^{n} g(k) \, P_{n-k}^{g,h}(x), \end{equation*} with initial value $P_{0}^{g,h}(x)=1$ encode information about several classical, widely studied polynomials. This includes Chebyshev polynomials of the second kind, associated Laguerre polynomials, and the Nekrasov--Okounkov polynomials. In this paper we prove that for $g(n)=n$ and fixed $h$ we obtain orthogonal polynomial sequences for positive definite functionals. Let $h(n)=n^s$ with $0 \leq s \leq 1 $. Then the sequence satisfies Tur\'an inequalities for $x \geq 0$.