Polynomials and algebraic curves related to certain binary and $b$-ary overpartitions

TL;DR

This paper explores polynomial sequences linked to binary and $b$-ary overpartitions, revealing their algebraic properties, zero distributions on algebraic curves, and connections to Chebyshev polynomials.

Contribution

It introduces new polynomial sequences related to overpartitions and studies their algebraic and combinatorial properties, including zero loci and extensions to $b$-ary cases.

Findings

Zeros of polynomial sequences lie on specific algebraic curves

Connections established between overpartition polynomials and Chebyshev polynomials

Extended analysis to $b$-ary overpartitions for any $b \\geq 2$

Abstract

We begin by considering a sequence of polynomials in three variables whose coefficients count restricted binary overpartitions with certain properties. We then concentrate on two specific subsequences that are closely related to the Chebyshev polynomials of both kinds, deriving combinatorial and algebraic properties of some special cases. We show that the zeros of these polynomial sequences lie on certain algebraic curves, some of which we study in greater detail. Finally, we extend part of this work to restricted $b$-ary overpartitions for arbitrary integers $b\geq 2$.