Kac--Moody Fibonacci sequences

TL;DR

The work develops a framework for generating infinite Fibonacci-type integer sequences from the root lattices of rank-2 Kac–Moody algebras. It constructs sequences $X_1(a,b)$, $X_2(a,b)$, and $X(k)$ from Weyl group actions and root coordinates, revealing linear recurrences and closed forms via Chebyshev polynomials, with key symmetries such as $X_1(a,b)=X_1(b,a)$ and $X_2(a,b)=a\,X(ab-2)$. By unifying these into a univariate $X(k)$ and providing explicit Chebyshev representations $x_{1,2m}(a,b)=U_{2m}(\sqrt{ab}/2)$ and $x_{2,2m-1}(a,b)=\sqrt{a/b}\,U_{2m-1}(\sqrt{ab}/2)$, the paper links a large class of Fibonacci-type sequences to root-lattice geometry. It complements the theory with comprehensive tabulations of the first twenty terms and cross-references to OEIS, highlighting structural regularities and diophantine connections that facilitate future exploration of these sequences in combinatorics and number theory.

Abstract

We summarize known results on how to generate an infinite family of integer sequences from the root lattices of rank 2 Kac--Moody algebras. We compute and tabulate the first twenty entries of a number of these sequences. This provides an overarching framework for a large class of Fibonacci-type integer sequences, evaluations of Chebyshev S and U-polynomials and others.

Figures (1)

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Theorems & Definitions (3)

• Lemma 1.1
• Theorem 2.1
• proof