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 $d_1\ne d_2$ of $n$, and integers $l_1$ and $l_2$ such that $0\le l_1\le \phi (d_1)-1$ and $0\le l_2\le \phi (d_2)-1$, we show that \[\left \langle X^{l_{1}} Ψ_{d_{1}}(X) (1+X^{d_1}+\dots X^{n-d_1}), X^{l_{2}} Ψ_{d_{2}}(X) (1+X^{d_2}+\dots X^{n-d_2}) \right \rangle =0, \] where $⟨\cdot ,\cdot ⟩$ is the inner product on $\mathbb{Q}[X]$ defined by $⟨{\sum }_k{a}_k{X}^k,{\sum }_k{b}_k{X}^k⟩={\sum }_k{a}_k{b}_k$.