Polynomial Continued Fractions for Algebraic Numbers

TL;DR

The paper proves that every real cubic algebraic number has a continued fraction expansion of polynomial type by combining GL$_2(\mathbb{Q})$-stability, Lagrange inversion-based hypergeometric representations of roots of cubic polynomials (notably $x^3-cx+c$ with large $|c|$), and Euler transformation to produce convergent polynomial-type CFs with convergents $p(n)/q(n)$ and asymptotics $u-p(n)/q(n)\sim C/(E^n n^{3/2})$ with $|E|$ arbitrarily large. It develops the notion of CF-type numbers ${\cal C}$, shows that cubic fields furnish such numbers, and provides explicit examples including $\root3\of{2}$ and roots of several discriminants. The work also discusses easy consequences, such as $u^a\in{\cal C}$ for rational $u>0$ and $a>0$, and outlines generalizations to higher degrees via similar Lagrange-inversion formulas, suggesting a broad applicability of polynomial-type CFs to algebraic numbers. This advances understanding of nonstandard continued fraction representations for cubic and higher-degree irrationals and opens avenues for further exploration in number theory and Diophantine approximation.

Abstract

Our main result is that any real cubic algebraic number has a continued fraction expansion with polynomial coefficients. Some generalizations are mentioned.

Theorems & Definitions (16)

• Definition 1.1
• Theorem 1.2
• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• Definition 2.4
• Lemma 2.5
• Lemma 2.6
• Corollary 2.7
• Lemma 2.8