Hyper $b$-ary expansions and Stern polynomials

TL;DR

This paper explores hyper $b$-ary expansions and Stern polynomials, generalizing known properties from base 2 to arbitrary bases, and introduces a matrix approach to analyze their number theoretic properties.

Contribution

It extends the theory of Stern polynomials to arbitrary bases and provides a matrix characterization, connecting to hyper $b$-ary partitions.

Findings

Generalized Stern polynomials to arbitrary bases

Developed a matrix characterization of Stern polynomials

Derived new results on hyper $b$-ary partitions

Abstract

We study a recently introduced base $b$ polynomial analog of Stern's diatomic sequence, which generalizes Stern polynomials of Klavar, Dilcher, Ericksen, Mansour, Stolarsky, and others. We lift some basic properties of base $2$ Stern polynomials to arbitrary base, and introduce a matrix characterization of Stern polynomials. By specializing, we recover some new number theoretic results about hyper $b$-ary partitions, which count partitions of $n$ into powers of $b$.