Divisibility and Arithmetic Properties of a Class of Sparse Polynomials

TL;DR

This paper explores the algebraic and arithmetic properties of a special class of sparse polynomials with binomial coefficients, revealing divisibility, irreducibility, and rational root characteristics, and studying related integer sequences.

Contribution

It introduces new results on divisibility, irreducibility, and rational roots of these polynomials, and analyzes their associated integer sequences and recurrence relations.

Findings

Identified divisibility properties of the polynomials

Proved irreducibility under certain conditions

Characterized rational roots and related integer sequences

Abstract

We investigate algebraic and arithmetic properties of a class of sequences of sparse polynomials that have binomial coefficients both as exponents and as coefficients. In addition to divisibility and irreducibility results we also consider rational roots. This leads to the study of an infinite class of integer sequences which have interesting properties and satisfy linear recurrence relations.