q-Analogue of Rational Numbers

• The q-analogue of a rational number is a canonical rational function in q that recovers the classical rational at q=1 and respects PSL(2,ℤ) symmetry.
• Its construction interweaves continued fractions, q-matrices, and combinatorial models to yield positive, unimodal polynomial coefficients.
• The theory connects algebraic, analytic, and combinatorial frameworks with applications in knot theory, representation theory, and q-calculus.

Searching arXiv for recent and foundational papers on q-deformed rationals and q-analogs of rational numbers. A q-analogue of a rational number, in the sense introduced by Morier-Genoud and Ovsienko, is a canonical rational function in the parameter $q$ attached to a classical rational $x\in\mathbb{Q}\cup\{\infty\}$. It is designed so that specialization at $q=1$ recovers $x$, while the deformation remains compatible with the modular action of $PSL(2,\mathbb{Z})$, continued fractions, and Farey geometry. For $x=n/m$ in lowest terms, one obtains

$$[x]_q=\frac{N(q)}{M(q)}, \qquad N(q),M(q)\in\mathbb{Z}[q],$$

with positive coefficients; this deformation is not the naive ratio $[n]_q/[m]_q$, but a genuinely two-variable construction in which numerator and denominator data are intertwined (Morier-Genoud et al., 31 Mar 2025).

1. Definition by modular invariance

The modern definition is formulated on the projective line $\mathbb{Q}\cup\{\infty\}$, where $PSL(2,\mathbb{Z})$ acts by fractional-linear transformations

$x\in\mathbb{Q}\cup\{\infty\}$0

Its $x\in\mathbb{Q}\cup\{\infty\}$1-deformed counterpart acts on $x\in\mathbb{Q}\cup\{\infty\}$2 through

$x\in\mathbb{Q}\cup\{\infty\}$3

and these satisfy the same defining relations as the classical generators: $x\in\mathbb{Q}\cup\{\infty\}$4 The $x\in\mathbb{Q}\cup\{\infty\}$5-rational associated to $x\in\mathbb{Q}\cup\{\infty\}$6 is the unique map $x\in\mathbb{Q}\cup\{\infty\}$7 characterized by $x\in\mathbb{Q}\cup\{\infty\}$8-equivariance,

$x\in\mathbb{Q}\cup\{\infty\}$9

together with the normalization $q=1$0 (Morier-Genoud et al., 31 Mar 2025).

Equivalent axiomatic recurrences are

$q=1$1

starting from $q=1$2. For rationals, these determine $q=1$3 uniquely and imply the auxiliary identities

$q=1$4

which make the modular character of the theory explicit (Leclere et al., 2021).

The specialization property is fundamental: $q=1$5 For $q=1$6 in lowest terms, the corresponding polynomials satisfy $q=1$7 and $q=1$8 (Morier-Genoud et al., 31 Mar 2025).

2. Continued fractions, matrices, and explicit formulas

Every rational admits two compatible continued-fraction descriptions, and each leads to a closed $q=1$9-formula. For the Hirzebruch, or minus-sign, continued fraction

$x$0

the deformation is obtained by replacing integers with Gaussian $x$1-integers and inserting powers of $x$2: $x$3 For the regular continued fraction

$x$4

one obtains an equivalent alternating expression involving both $x$5 and $x$6 (Morier-Genoud et al., 31 Mar 2025).

These constructions admit a convergent recursion. If

$x$7

then

$x$8

The same rational function is also obtained from $x$9 $PSL(2,\mathbb{Z})$0-matrices such as

$PSL(2,\mathbb{Z})$1

or, in an equivalent presentation,

$PSL(2,\mathbb{Z})$2

which encode the continued-fraction word directly (Morier-Genoud et al., 31 Mar 2025, Aval et al., 14 Nov 2025).

Typical values are as follows.

$PSL(2,\mathbb{Z})$3	Continued fraction	$PSL(2,\mathbb{Z})$4
$PSL(2,\mathbb{Z})$5	$PSL(2,\mathbb{Z})$6	$PSL(2,\mathbb{Z})$7
$PSL(2,\mathbb{Z})$8	$PSL(2,\mathbb{Z})$9	$x=n/m$0
$x=n/m$1	$x=n/m$2	$x=n/m$3
$x=n/m$4	$x=n/m$5	$x=n/m$6

These examples illustrate the general phenomenon that both numerator and denominator have nonnegative coefficients and that evaluation at $x=n/m$7 reproduces the original fraction (Morier-Genoud et al., 31 Mar 2025).

3. Positivity, total positivity, and symmetry

A central structural feature is positivity. For every rational $x=n/m$8, the polynomials $x=n/m$9 and $$[x]_q=\frac{N(q)}{M(q)}, \qquad N(q),M(q)\in\mathbb{Z}[q],$$0 in

$$[x]_q=\frac{N(q)}{M(q)}, \qquad N(q),M(q)\in\mathbb{Z}[q],$$1

have positive integer coefficients (Morier-Genoud et al., 31 Mar 2025). Earlier formulations established the same positivity through continued-fraction recurrences, triangulations, and quiver models (Morier-Genoud et al., 2018).

A stronger statement is total positivity. If $$[x]_q=\frac{N(q)}{M(q)}, \qquad N(q),M(q)\in\mathbb{Z}[q],$$2 and

$$[x]_q=\frac{N(q)}{M(q)}, \qquad N(q),M(q)\in\mathbb{Z}[q],$$3

then

$$[x]_q=\frac{N(q)}{M(q)}, \qquad N(q),M(q)\in\mathbb{Z}[q],$$4

has positive integer coefficients. Moreover,

$$[x]_q=\frac{N(q)}{M(q)}, \qquad N(q),M(q)\in\mathbb{Z}[q],$$5

so Farey-neighbor relations are detected exactly by monomials (Morier-Genoud et al., 31 Mar 2025).

For successive convergents one has the determinant identity

$$[x]_q=\frac{N(q)}{M(q)}, \qquad N(q),M(q)\in\mathbb{Z}[q],$$6

the $$[x]_q=\frac{N(q)}{M(q)}, \qquad N(q),M(q)\in\mathbb{Z}[q],$$7-analogue of the classical continuant Wronskian formula (Morier-Genoud et al., 2019). Matrix traces exhibit an additional symmetry: for any $$[x]_q=\frac{N(q)}{M(q)}, \qquad N(q),M(q)\in\mathbb{Z}[q],$$8, the trace of $$[x]_q=\frac{N(q)}{M(q)}, \qquad N(q),M(q)\in\mathbb{Z}[q],$$9 is palindromic (Morier-Genoud et al., 31 Mar 2025).

Unimodality is another prominent property. For every rational $[n]_q/[m]_q$0, the coefficient sequences of $[n]_q/[m]_q$1 and $[n]_q/[m]_q$2 are unimodal, extending the classical Sylvester–Cayley phenomenon for Gaussian $[n]_q/[m]_q$3-binomials (Morier-Genoud et al., 31 Mar 2025). A later combinatorial approach also proves unimodality through rank polynomials of fence posets (Aval et al., 14 Nov 2025).

4. Farey, snake graphs, posets, quivers, and Schubert geometry

The theory has several equivalent combinatorial realizations. On the Farey graph, the ordinary mediant operation is replaced by a $[n]_q/[m]_q$4-weighted mediant

$[n]_q/[m]_q$5

with the exponent determined inductively from the initial triangle $[n]_q/[m]_q$6. This recursion reproduces exactly the $[n]_q/[m]_q$7-rationals across the Farey tessellation and is the analogue of the weighted Pascal identity for Gaussian $[n]_q/[m]_q$8-binomial coefficients (Morier-Genoud et al., 31 Mar 2025).

In the original construction, a rational $[n]_q/[m]_q$9 is encoded by a triangulation of a polygon. The coefficients of the numerator and denominator then count closure subsets of oriented $\mathbb{Q}\cup\{\infty\}$0-type quivers dual to that triangulation; equivalently, they count subrepresentations of the maximal indecomposable representation of the corresponding quiver (Morier-Genoud et al., 2018).

A second family of models uses snake graphs. If $\mathbb{Q}\cup\{\infty\}$1, its snake graph is a zig-zag polyomino with $\mathbb{Q}\cup\{\infty\}$2 unit squares. The coefficient of $\mathbb{Q}\cup\{\infty\}$3 in the numerator polynomial $\mathbb{Q}\cup\{\infty\}$4 equals the number of NE-lattice paths in the snake graph with $\mathbb{Q}\cup\{\infty\}$5 boxes below. The denominator has an analogous interpretation in a smaller snake graph (Morier-Genoud et al., 31 Mar 2025). The same numerator polynomial also appears as a perfect-matching generating function, via the specialization of cluster-algebra $\mathbb{Q}\cup\{\infty\}$6-polynomials (Ovenhouse, 2021).

Fence posets provide an equivalent order-theoretic description. For an even-length regular continued fraction, one constructs a fence poset whose lattice of lower ideals has rank-generating function equal to $\mathbb{Q}\cup\{\infty\}$7 (Ovenhouse, 2021). A later refinement treats all positive rationals, including those in $\mathbb{Q}\cup\{\infty\}$8, by using a single fence poset or a single snake graph together with an internal partition that simultaneously yields numerator and denominator generating functions (Aval et al., 14 Nov 2025).

The geometric interpretation is formulated over finite fields. For a snake graph identified with a skew Young diagram $\mathbb{Q}\cup\{\infty\}$9, one has

$PSL(2,\mathbb{Z})$0

where the union ranges over open Schubert cells in a Grassmannian $PSL(2,\mathbb{Z})$1. Thus the numerator counts $PSL(2,\mathbb{Z})$2-points in a union of open Schubert cells, paralleling the classical finite-field interpretation of Gaussian $PSL(2,\mathbb{Z})$3-binomials (Ovenhouse, 2021).

5. Extension to irrationals and analytic theory

The rational theory extends to real numbers by stabilization. If $PSL(2,\mathbb{Z})$4 is irrational and $PSL(2,\mathbb{Z})$5 is any rational sequence, then the Taylor expansions

$PSL(2,\mathbb{Z})$6

stabilize coefficientwise: for each fixed $PSL(2,\mathbb{Z})$7, the coefficient $PSL(2,\mathbb{Z})$8 is eventually constant. The limit defines a formal power series

$PSL(2,\mathbb{Z})$9

independent of the approximating sequence (Morier-Genoud et al., 31 Mar 2025).

If

$x\in\mathbb{Q}\cup\{\infty\}$00

is the infinite regular continued fraction of an irrational, then $x\in\mathbb{Q}\cup\{\infty\}$01 is given by the corresponding infinite $x\in\mathbb{Q}\cup\{\infty\}$02-continued fraction, and truncation after $x\in\mathbb{Q}\cup\{\infty\}$03 terms stabilizes at least the first $x\in\mathbb{Q}\cup\{\infty\}$04 Taylor coefficients (Morier-Genoud et al., 31 Mar 2025). For quadratic irrationals, $x\in\mathbb{Q}\cup\{\infty\}$05 often has a closed radical form

$x\in\mathbb{Q}\cup\{\infty\}$06

with $x\in\mathbb{Q}\cup\{\infty\}$07 palindromic. For the golden ratio,

$x\in\mathbb{Q}\cup\{\infty\}$08

(Morier-Genoud et al., 31 Mar 2025).

The analytic question is the radius of convergence $x\in\mathbb{Q}\cup\{\infty\}$09 when $x\in\mathbb{Q}\cup\{\infty\}$10 is complex. A universal lower bound holds: $x\in\mathbb{Q}\cup\{\infty\}$11 A conjecturally optimal bound is

$x\in\mathbb{Q}\cup\{\infty\}$12

with equality expected precisely for numbers $x\in\mathbb{Q}\cup\{\infty\}$13-equivalent to the golden ratio; this is the proposed $x\in\mathbb{Q}\cup\{\infty\}$14-analogue of Hurwitz’s irrational number theorem (Leclere et al., 2021). For rational classes whose Hirzebruch–Jung coefficients satisfy $x\in\mathbb{Q}\cup\{\infty\}$15 eventually, the stronger inequality $x\in\mathbb{Q}\cup\{\infty\}$16 is proved (Leclere et al., 2021).

At rational points, stabilization depends on the side of approximation. Besides the “right” value $x\in\mathbb{Q}\cup\{\infty\}$17, there is a “left” value

$x\in\mathbb{Q}\cup\{\infty\}$18

which is independent of the chosen $x\in\mathbb{Q}\cup\{\infty\}$19. Approximations from below stabilize to $x\in\mathbb{Q}\cup\{\infty\}$20, while approximations from above stabilize to $x\in\mathbb{Q}\cup\{\infty\}$21. The two are intertwined by an involution $x\in\mathbb{Q}\cup\{\infty\}$22 (Morier-Genoud et al., 31 Mar 2025).

6. Relations to knot theory, representation theory, and $x\in\mathbb{Q}\cup\{\infty\}$23-calculus

The construction is part of the broader theory of $x\in\mathbb{Q}\cup\{\infty\}$24-deformation. Its modular $x\in\mathbb{Q}\cup\{\infty\}$25-action places it near $x\in\mathbb{Q}\cup\{\infty\}$26-calculus and the representation-theoretic context in which $x\in\mathbb{Q}\cup\{\infty\}$27-deformations underlie quantum groups (Morier-Genoud et al., 31 Mar 2025). The basic recursion

$x\in\mathbb{Q}\cup\{\infty\}$28

is the simplest manifestation of this connection.

For rational knots, or $x\in\mathbb{Q}\cup\{\infty\}$29-bridge knots in Conway’s parametrization, the normalized Jones polynomial is expressed directly in terms of the numerator and denominator polynomials of the corresponding $x\in\mathbb{Q}\cup\{\infty\}$30-rational. In one formulation,

$x\in\mathbb{Q}\cup\{\infty\}$31

so the knot invariant is a linear combination of the two constituents of $x\in\mathbb{Q}\cup\{\infty\}$32 (Morier-Genoud et al., 2018).

Representation theory and Schubert calculus enter through the finite-field Grassmannian interpretation of the numerator, while cluster algebras enter through snake graphs, perfect matchings, and $x\in\mathbb{Q}\cup\{\infty\}$33-polynomials (Ovenhouse, 2021). Discrete integrable systems also appear: in quadratic cases, the coefficients of $x\in\mathbb{Q}\cup\{\infty\}$34 give rise to Somos-type recurrences for associated Hankel determinants (Morier-Genoud et al., 31 Mar 2025).

A distinct arithmetic connection is provided by the $x\in\mathbb{Q}\cup\{\infty\}$35-analog of Markoff numbers. Using Christoffel words and a matrix morphism $x\in\mathbb{Q}\cup\{\infty\}$36, one obtains a perfect-matching generating formula whose specialization at $x\in\mathbb{Q}\cup\{\infty\}$37 recovers the classical Markoff number (Aval et al., 14 Nov 2025).

7. Comparisons, misconceptions, and open problems

A persistent misconception is to identify the $x\in\mathbb{Q}\cup\{\infty\}$38-analogue of $x\in\mathbb{Q}\cup\{\infty\}$39 with the naive quotient $x\in\mathbb{Q}\cup\{\infty\}$40. That quotient does recover $x\in\mathbb{Q}\cup\{\infty\}$41 at $x\in\mathbb{Q}\cup\{\infty\}$42, but it does not satisfy the modular invariance, does not encode the continued-fraction geometry, and has no refined combinatorial or finite-field interpretation. By contrast, the Morier-Genoud–Ovsienko deformation is $x\in\mathbb{Q}\cup\{\infty\}$43-equivariant, positive, unimodal, and compatible with Farey and Schubert structures (Morier-Genoud et al., 31 Mar 2025).

Several questions remain open. The main analytic problem is the $x\in\mathbb{Q}\cup\{\infty\}$44-Hurwitz conjecture asserting that

$x\in\mathbb{Q}\cup\{\infty\}$45

for all real $x\in\mathbb{Q}\cup\{\infty\}$46, with equality exactly on the $x\in\mathbb{Q}\cup\{\infty\}$47-orbit of the golden ratio (Leclere et al., 2021). Other open directions include sharper lower bounds for rational radii of convergence, the characterization of which integer-coefficient power series arise as $x\in\mathbb{Q}\cup\{\infty\}$48-deformed reals, the algebraicity of radii for general quadratic irrationals, and finer zero-distribution results for Fibonacci and Pell polynomial families associated with convergents (Leclere et al., 2021).

On the combinatorial side, recent work shows that a single fence poset or a single snake graph can generate both numerator and denominator for every positive rational, extending earlier interpretations that were restricted to rationals $x\in\mathbb{Q}\cup\{\infty\}$49 or required separate objects (Aval et al., 14 Nov 2025). This suggests that the topic is still being reorganized around a more unified combinatorial infrastructure, while the modular and analytic aspects continue to develop in parallel.