On certain arithmetic properties of Stern polynomials

TL;DR

This paper investigates the arithmetic properties of Stern polynomials, including their degree sequence, providing new theorems and insights into their structure and behavior.

Contribution

It introduces several new theorems about the arithmetic properties and degree sequence of Stern polynomials, expanding understanding of their algebraic structure.

Findings

Derived new theorems on Stern polynomial properties

Analyzed the degree sequence e(n) and its characteristics

Provided insights into the algebraic and combinatorial structure of Stern polynomials

Abstract

We prove several theorems concerning arithmetic properties of Stern polynomials defined in the following way: $B_{0}(t)=0, B_{1}(t)=1, B_{2n}(t)=tB_{n}(t)$, and $B_{2n+1}(t)=B_{n}(t)+B_{n+1}(t)$. We study also the sequence $e(n)=\op{deg}_{t}B_{n}(t)$ and give various of its properties.