An orthogonal relation on inverse 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}$.