q-Rational Numbers: Theory and Applications

• q-Rational Numbers are q-deformations of classical rationals, defined via q-integers and q-continued fractions that generalize continued fraction expansions.
• They exhibit rich algebraic and combinatorial structures, including positive, unimodal, and palindromic polynomial recurrences linked to cluster algebras, quiver theory, and knot invariants.
• Their analytic and geometric features are underscored by explicit radii of convergence, q-Ford circles, and deformed Farey tessellations, providing deep insights into modular and combinatorial properties.

A $q$-rational number is a $q$-deformation of the classical notion of a rational number, introduced within the theory of quantizations, combinatorics, knot invariants, and geometric representation theory. The subject arises from the systematic extension of $q$-integers to rational domains, generalizing continued fractions, Farey geometry, and the algebraic and combinatorial structures underlying them. Developed primarily by Morier-Genoud and Ovsienko, $q$-rational numbers encompass canonical $q$-deformations of both rationals and irrationals, with algebraic, geometric, combinatorial, and representation-theoretic interpretations and deep connections to cluster algebras, quiver theory, and knot invariants (Morier-Genoud et al., 2018, Morier-Genoud et al., 31 Mar 2025).

1. Definitions and Core Constructions

Let $q$ be a formal parameter. For $n\geq 0$, define the $q$-integer

$$[n]_q = \frac{1 - q^n}{1 - q} = 1 + q + \cdots + q^{n-1} \in \mathbb{Z}[q].$$

Given a rational number $x=\frac{r}{s}>0$ in lowest terms, write its even-length continued fraction expansion

$q$0

and define its $q$1-rational by a nested $q$2-continued fraction: $q$3 Similar constructions use negative (Hirzebruch-type) continued fractions, yielding equivalent results. For every rational $q$4,

$q$5

Alternatively, $q$6-rationals can be defined via a $q$7-deformation of the modular action: $q$8 which, with appropriate normalization, yields a unique $q$9-equivariant $q$0-rational map $q$1 with $q$2 (Morier-Genoud et al., 31 Mar 2025, Morier-Genoud et al., 2018, Aval et al., 14 Nov 2025).

2. $q$3-Rationals, Irrationals, and Quadratic Cases

For $q$4, one defines $q$5 by the coefficientwise stabilization of the power series expansion of $q$6 for any rational sequence $q$7. The resulting $q$8 is independent of the choice of approximants (Ren, 2021, Morier-Genoud et al., 31 Mar 2025).

In the case of (right) $q$9-deformed quadratic irrationals, for periodic continued fractions $q$0 (classical metallic numbers), one has

$q$1

which is the limit of the $q$2-rationals of the truncated expansions. These satisfy explicit algebraic equations: for the $q$3th $q$4-metallic number, $q$5 is a root of

$q$6

with discriminant $q$7 a palindromic polynomial of degree $q$8 (Ren, 2021). For general quadratic irrationals, $q$9 is algebraic of degree $q$0, with explicit polynomial formulas (Morier-Genoud et al., 31 Mar 2025).

3. Structural Properties and Recurrences

The $q$1-rationals satisfy the $q$2-equivariance

$q$3

together with fundamental recursions: $q$4 For every continued fraction $q$5, the associated $q$6 polynomials satisfy three-term or four-term continuant recurrences, directly generalizing classical continuants and tridiagonal determinant formulas: $q$7 yielding numerator and denominator polynomials with positive integer coefficients (Morier-Genoud et al., 31 Mar 2025, Morier-Genoud et al., 2018, Ren, 2021, Aval et al., 14 Nov 2025).

These recurrences also extend to $q$8-metallic numbers via interlaced $q$9-rational sequences $q$0, generalizing classical integer and Pell/Fibonacci recursions (Ren, 2021).

4. Positivity, Total Positivity, and Combinatorics

A central property is total positivity: for rationals $q$1, with $q$2 and $q$3,

$q$4

The coefficients of $q$5 are unimodal and, for palindromic continued fractions, palindromic (Morier-Genoud et al., 2018, Morier-Genoud et al., 31 Mar 2025).

Multiple combinatorial and geometric interpretations exist:

• The coefficients of $q$6 enumerate north-east lattice paths, order ideals in fence posets, perfect matchings in snake graphs, and certain $q$7-partitions in skew shapes (Aval et al., 14 Nov 2025, Ovenhouse, 2021, Burcroff et al., 2024).
• For $q$8 a prime power, these numbers count $q$9-points in explicit unions of Schubert cells of finite Grassmannians, providing a geometric realization (Ovenhouse, 2021).
• The $n\geq 0$0-rational numerators for certain Christoffel words recover $n\geq 0$1-Markoff numbers, connected to perfect matchings in snake graphs (Aval et al., 14 Nov 2025).

5. Analytic and Algebraic Features: Radii of Convergence

The power series expansion $n\geq 0$2 has a radius of convergence determined by the nearest zero of a characteristic polynomial. For quadratic irrationals and, in particular, $n\geq 0$3-metallic numbers, the singularities are zeros of the discriminant $n\geq 0$4, a palindromic polynomial; all such zeros have modulus in an explicit annulus: $n\geq 0$5 A universal analytic lower bound holds: for all real $n\geq 0$6, the radius of convergence of $n\geq 0$7 is at least $n\geq 0$8, with this minimal value achieved precisely for the modular orbit of the golden ratio. For finite $n\geq 0$9-rational approximants (truncated expansions), the radius of convergence is strictly larger for $q$0 (Ren, 2021, Morier-Genoud et al., 31 Mar 2025).

6. Geometric and Categorical Aspects

Every $q$1-rational number $q$2 can be geometrically represented as a circle in the complex plane—the $q$3-Ford circle—generalizing the classical Ford circles. The $q$4-deformed Farey tessellation and modular orbifold are natural generalizations, with the length and area of the modular surface explicitly computable as functions of $q$5 (Jouteur et al., 4 Mar 2026). The compactification of spaces of Bridgeland stability conditions for 2-Calabi–Yau categories (of type $q$6) realizes $q$7-rationals as boundary points, with braid group ($q$8) orbits governing their symmetries (Bapat et al., 2022).

Quadratic operations (Springborn sums) between $q$9-rationals correspond to algebraic combinations of $$[n]_q = \frac{1 - q^n}{1 - q} = 1 + q + \cdots + q^{n-1} \in \mathbb{Z}[q].$$0-Ford circle centers, reducing to $$[n]_q = \frac{1 - q^n}{1 - q} = 1 + q + \cdots + q^{n-1} \in \mathbb{Z}[q].$$1-mediants in regular cases and satisfying deformed Markov-type equations in iterated instances (Jouteur et al., 4 Mar 2026).

7. Applications and Further Directions

$$[n]_q = \frac{1 - q^n}{1 - q} = 1 + q + \cdots + q^{n-1} \in \mathbb{Z}[q].$$2-rational numbers have ramifications across several domains:

• The numerator/denominator polynomials $$[n]_q = \frac{1 - q^n}{1 - q} = 1 + q + \cdots + q^{n-1} \in \mathbb{Z}[q].$$3 appear in the explicit formulas for Jones polynomials of rational knots; $$[n]_q = \frac{1 - q^n}{1 - q} = 1 + q + \cdots + q^{n-1} \in \mathbb{Z}[q].$$4-binomial and $$[n]_q = \frac{1 - q^n}{1 - q} = 1 + q + \cdots + q^{n-1} \in \mathbb{Z}[q].$$5-gamma identities involving $$[n]_q = \frac{1 - q^n}{1 - q} = 1 + q + \cdots + q^{n-1} \in \mathbb{Z}[q].$$6-rationals yield generalizations of the classical hypergeometric series, combinatorial identities, and $$[n]_q = \frac{1 - q^n}{1 - q} = 1 + q + \cdots + q^{n-1} \in \mathbb{Z}[q].$$7-Gamma function reflection formulas (Machacek et al., 2023, Morier-Genoud et al., 2018).
• For $$[n]_q = \frac{1 - q^n}{1 - q} = 1 + q + \cdots + q^{n-1} \in \mathbb{Z}[q].$$8, higher $$[n]_q = \frac{1 - q^n}{1 - q} = 1 + q + \cdots + q^{n-1} \in \mathbb{Z}[q].$$9-continued fractions generalize $x=\frac{r}{s}>0$0-rational numbers, with analogous combinatorial and positivity properties (Burcroff et al., 2024).
• The algebraic framework supports links to cluster algebras (as $x=\frac{r}{s}>0$1-polynomials), quiver representations, and dimer models on snake graphs, thereby providing a unification of enumerative, geometric, and algebraic combinatorics (Ovsienko, 17 Oct 2025).
• Future directions include the further study of $x=\frac{r}{s}>0$2-deformed Diophantine structures (Markov numbers), $x=\frac{r}{s}>0$3-integration in representation theory, and connections with modular retrieval in analytic number theory and quantum topology (Aval et al., 14 Nov 2025, Morier-Genoud et al., 31 Mar 2025).