Pairs of intertwined integer sequences

Abstract

In previous work we computed the number $C_n(q)$ of ideals of codimension $n$ of the algebra ${\mathbb{F}}_q[x,y,x^{-1},y^{-1}]$ of two-variable Laurent polynomials over a finite field: it turned out that $C_n(q)$ is a palindromic polynomial of degree $2n$ in $q$, divisible by $(q-1{)}^2$. The quotient $P_n(q)=C_n(q)/(q-1{)}^2$ is a palindromic polynomial of degree $2n-2$. For each $n\ge 1$ let ${\overline{P}}_n(X)\in \mathbb{Z}[X]$ be the degree $n-1$ polynomial such that ${\overline{P}}_n(q+q^{-1})=P_n(q)/q^{n-1}$. In this note we show that for any integer $N$ the integer value ${\overline{P}}_n(N)$ is close to the value at $N$ of the degree $n-1$ polynomial $F_{n-1}(X)=1+{\sum }_{k=1}^{n-1}{\overline{T}}_k(X)$, which is a sum of monic versions ${\overline{T}}_k(X)$ of Chebyshev polynomials of the first kind. We give a precise formula for ${\overline{P}}_n(X)$ as a linear combination of $F_k(X)$'s, each appearance of the latter being parametrized by an odd divisor of $n$. As a consequence, ${\overline{P}}_n(X)=F_{n-1}(X)$ if and only if $n$ is a power of $2$. We exhibit similar formulas for $C_n(q)$.

20 pages. Version 2: Propositions 2.1 and 3.1 added; one reference added and Remark 4.7 modified