Some Properties of a Class of Sparse Polynomials

TL;DR

This paper investigates a broad class of sparse polynomials with binomial coefficients as exponents and coefficients, exploring their identities, properties, and bounds on zeros, extending previous graph-theoretic polynomial results.

Contribution

It introduces a generalized class of sparse polynomials with binomial coefficients, deriving new identities and bounds on zeros, expanding understanding beyond specific graph-related cases.

Findings

Established identities involving these polynomials

Proved monotonicity and log-concavity properties

Derived bounds on the zeros' moduli

Abstract

We study an infinite class of sequences of sparse polynomials that have binomial coefficients both as exponents and as coefficients. This generalizes a sequence of sparse polynomials which arises in a natural way as graph theoretic polynomials. After deriving some basic identities, we obtain properties concerning monotonicity and log-concavity, as well as identities involving derivatives. We also prove upper and lower bounds on the moduli of the zeros of these polynomials.