Zeros of recursively defined polynomials

Abstract

Let g, h be two arithmetic functions. In this paper, we study zero-free intervals of polynomials of the type Png,h(x):=xh(n)∑k=1ng(k)Pn−kg,h(x),P0g,h(x)=1. We use a difference equation approach. Let g(n)=σ(n) be the divisor sum function and h(n) the identity function, then the zeros of the polynomials dictate the vanishing properties of the coefficients of all powers of the Dedekind eta function. This is directly related to the Lehmer conjecture on the non-vanishing of the Ramanujan τ-function. Let g(n) be the identity function. Then, the polynomials are related to generalized Laguerre polynomials for h(n)=n and to Chebyshev polynomials of the second kind for h(n)=1. The paper is furnished with several interesting examples. We found polynomials with decreasing minimal real zeros. We provide a simple arithmetic function g to approximate the minimal real zeros attached to σ.