Turán inequalities from Chebyshev to Laguerre polynomials

Abstract

Let g and h be real-valued arithmetic functions, positive and normalized. Specific choices within the following general scheme of recursively defined polynomialswith initial value P g,h 0 (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 = id and fixed h we obtain orthogonal polynomial sequences for positive definite functionals. Let h(n) = n s with 0 ≤ s ≤ 1. Then the sequence satisfies Turán inequalities for x ≥ 0.