标题： 交织整数序列对 显示英文标题

标题： Pairs of intertwined integer sequences

摘要： 在以前的工作中，我们计算了有限域上二元 Laurent 多项式代数 ${\mathbb{F}}_q[x,y,x^{-1}, y^{-1}]$ 的余维数为 $n$的理想的数量 $C_n(q)$：结果表明， $C_n(q)$是一个次数为 $2n$的关于 $q$的回文多项式，并且可被 $(q-1)^2$整除。 商式$P_n(q) = C_n(q)/(q-1)^2$是一个次数为$2n-2$的回文多项式。 对于每个$n\geq 1$，存在一个唯一的次数为$n-1$的多项式${\overline{P}}_n(X) \in {\mathbb{Z}}[X]$，使得${\overline{P}}_n(q+q^{-1}) = P_n(q)/q^{n-1}$。 在本文中，我们证明对于任何整数$N$，整数值${\overline{P}}_n(N)$接近于次数为$n-1$的多项式$F_{n-1}(X) = 1 + \sum_{k=1}^{n-1} \, {\overline{T}}_k(X)$在$N$处的值，该多项式是第一类切比雪夫多项式的首一形式${\overline{T}}_k(X)$的和。 我们给出${\overline{P}}_n(X)$的精确公式，作为$F_k(X)$的线性组合，每个后者出现的情况由$n$的一个奇数因子参数化。 作为推论，${\overline{P}}_n(X) = F_{n-1}(X)$当且仅当$n$是$2$的幂。 我们为$C_n(q)$展示类似的公式。 显示英文摘要

摘要： 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 $n\geq 1$ there is a unique degree $n-1$ polynomial ${\overline{P}}_n(X) \in {\mathbb{Z}}[X]$ 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 $C_n(q)$.