A study of a recursive sequence of polynomials revealing weighted Catalan Numbers

TL;DR

This work studies the recursive polynomial sequence $p_n$ defined by $p_0(x)=x^2-2$ and $p_n(x)=p_{n-1}(x)^2-2$, motivated by field-theoretic constructions related to cyclotomic generators. It derives a recursive coefficient formula for the even coefficients $c_{n,2k}$ via minimal-polynomial considerations and uses a Vandermonde framework to extract invariants $a_{j,k}$, including a diagonal invariant $a_{k,k}$. A key result is that $a_{k,k}$ admits a combinatorial, weighted-Catalan representation through labeled ordered trees, connecting algebraic and enumerative structures. Overall, the paper unveils a bridge between cyclotomic-field minimal polynomials and Catalan-type combinatorics, and provides computational methods for the coefficients and invariants of the polynomial sequence.

Abstract

This paper examines the recursive sequence of polynomials $p_n(x)$, defined by $p_0(x) = x^2 - 2$ and $p_n(x) = p_{n-1}(x)^2 - 2$ for $n \geq 1$. It describes the field-theoretic motivations behind this sequence, derives a recursive formula for its coefficients, and identifies invariants that uncover combinatorial connections, including links to weighted Catalan numbers.

Theorems & Definitions (30)

• Definition 1.1: Lemma
• Remark 1.2
• Example 1.3
• Definition 1.4
• Proposition 1.5
• proof
• Remark 1.6
• Corollary 1.7
• proof
• Theorem 2.1