An orthogonal relation on inverse cyclotomic polynomials

TL;DR

This paper establishes an orthogonal relation between certain inverse cyclotomic polynomial-based polynomials, revealing their inner product is zero for distinct divisor pairs, contributing to the understanding of their algebraic structure.

Contribution

It introduces a novel orthogonality relation for inverse cyclotomic polynomials associated with different divisors, expanding the theoretical framework of cyclotomic polynomial interactions.

Findings

Proves orthogonality of specific inverse cyclotomic polynomial combinations

Shows zero inner product for polynomials with different divisor indices

Enhances understanding of algebraic relations among cyclotomic polynomials

Abstract

Let $\Phi_n(X)$ and $\Psi_n(X)=\frac{X^{n}-1}{\Phi_{n}(X)}$ be the $n$-th cyclotomic and inverse cyclotomic polynomials respectively. In this short note, for any pair of divisors $ d_{1} \neq d_{2} $ of $ n $, and integers $l_1$ and $l_2$ such that $ 0 \leq l_{1} \leq \varphi(d_{1})-1 $ and $ 0 \leq l_{2} \leq \varphi(d_{2})-1 $, we show that \[\left \langle X^{l_{1}} \Psi_{d_{1}}(X) (1+X^{d_1}+\dots X^{n-d_1}), X^{l_{2}} \Psi_{d_{2}}(X) (1+X^{d_2}+\dots X^{n-d_2}) \right \rangle =0, \] where $ \langle \cdot, \cdot \rangle $ is the inner product on $\mathbb{Q}[X]$ defined by $ \langle \sum_{k} a_{k}X^{k},\sum_{k} b_{k}X^{k} \rangle =\sum_{k} a_{k}b_{k}$.