A digit reversal property for Stern polynomials

TL;DR

This paper introduces a polynomial generalization of Stern's diatomic series and proves a digit reversal symmetry property, revealing a novel invariance under binary digit reversal.

Contribution

It establishes a new digit reversal invariance property for a generalized Stern polynomial sequence, extending known symmetries in binary representations.

Findings

Polynomials are invariant under binary digit reversal.

Coefficient interpretation relates to hyperbinary expansions.

Proves a new symmetry property of Stern polynomials.

Abstract

We consider the following polynomial generalization of Stern's diatomic series: let $s_1(x,y)=1$, and for $n\geq 1$ set $s_{2n}(x,y)=s_n(x,y)$ and $s_{2n+1}(x,y)=x\,s_n(x,y)+y\,s_{n+1}(x,y)$. The coefficient $[x^iy^j]s_n(x,y)$ is the number of hyperbinary expansions of $n-1$ with exactly $i$ occurrences of the digit $\mathtt 2$ and $j$ occurrences of $\mathtt 0$. We prove that the polynomials $s_n$ are invariant under \emph{digit reversal}, that is, $s_n=s_{n^R}$, where $n^R$ is obtained from $n$ by reversing the binary expansion of $n$.