q-Deformed Gaussian Integers

• q-Deformed Gaussian integers are a q-analog of classical Gaussian integers, constructed via modular-equivariant deformations of PSL(2, ℤ) actions.
• They feature explicit closed-form expressions using Euler q-integers and Chebyshev polynomial recurrences to capture their additive structure.
• While they lack a natural deformed multiplicative law, their unique construction recovers the classical Gaussian integers as q approaches 1.

$q$-deformed Gaussian integers are a modular-equivariant $q$-analog of the classical Gaussian integers $\mathbb{Z}[i]$, constructed to preserve the symmetry of the modular group $\mathrm{PSL}(2, \mathbb{Z})$ under $q$-deformation. These objects generalize both Euler $q$-integers and the lattice of Gaussian integers, while encoding deep group-theoretic and combinatorial structures, including explicit connections to Chebyshev polynomials. Their theory is characterized by uniqueness, closed-form expressions, and an explicit additive structure, but notably lacks a natural deformation of multiplication (Ovsienko, 2021).

1. $q$-Deformed Complex Numbers and Modular Invariance

Let $\mathbb{Q}$ denote the rationals, and $\mathbb{C}(q^2)$ the field of rational functions in the formal parameter $q^2$. The construction relies on two actions:

• The standard $q$0-deformations $q$1 and $q$2 of the generators of $q$3, acting by linear-fractional transformations on $q$4:

$q$5

which preserve $q$6's relations: $q$7.

• The involutive automorphism $q$8.

A $q$9-deformation of $\mathbb{Z}[i]$0 is a map $\mathbb{Z}[i]$1, uniquely characterized by $\mathbb{Z}[i]$2 and

$\mathbb{Z}[i]$3

ensuring modular equivariance. The construction recovers the Euler $\mathbb{Z}[i]$4-integers as

$\mathbb{Z}[i]$5

which underpins the additive $\mathbb{Z}[i]$6-arithmetic for deformed Gaussian integers.

2. The $\mathbb{Z}[i]$7-Analog of Translation by $\mathbb{Z}[i]$8

Classically, $\mathbb{Z}[i]$9 is the orbit of $\mathrm{PSL}(2, \mathbb{Z})$0 under translations $\mathrm{PSL}(2, \mathbb{Z})$1 and $\mathrm{PSL}(2, \mathbb{Z})$2 in $\mathrm{PSL}(2, \mathbb{Z})$3. The imaginary unit $\mathrm{PSL}(2, \mathbb{Z})$4 is fixed by $\mathrm{PSL}(2, \mathbb{Z})$5 ($\mathrm{PSL}(2, \mathbb{Z})$6), leading to $\mathrm{PSL}(2, \mathbb{Z})$7 and $\mathrm{PSL}(2, \mathbb{Z})$8 from $\mathrm{PSL}(2, \mathbb{Z})$9. To define the $q$0-analog $q$1 of $q$2, the required properties are:

• $q$3 commutes with $q$4,
• $q$5, $q$6.

No genuine matrix in $q$7 meets these, but a unique twisted linear-fractional map does:

$q$8

which satisfies

$q$9

The square $q$0 yields the $q$1-deformed two-step translation as a specific matrix linear-fractional transformation.

3. Definition and Structure of $q$2-Deformed Gaussian Integers

Define an action of $q$3 on $q$4 by independent application of $q$5 and $q$6. The $q$7-deformed Gaussian integers are

$q$8

This realizes $q$9-Gaussian integers as the $q$0-translated orbit of $q$1 in $q$2. These satisfy the closed form

$q$3

where $q$4 is the Euler $q$5-integer, and $q$6 is further decomposed.

4. Closed Form via a Second Deformation Parameter $q$7

Let

$q$8

The imaginary components $q$9 are given by

$\mathbb{Q}$0

with $\mathbb{Q}$1 recursively determined. Explicitly,

• For $\mathbb{Q}$2,

$\mathbb{Q}$3

with $\mathbb{Q}$4 and $\mathbb{Q}$5.

• For $\mathbb{Q}$6,

$\mathbb{Q}$7

Thus, all $\mathbb{Q}$8-Gaussian integers admit rational expressions in $\mathbb{Q}$9 and $\mathbb{C}(q^2)$0.

5. Connection with Chebyshev Polynomials

Setting

$\mathbb{C}(q^2)$1

one defines

$\mathbb{C}(q^2)$2

The sequences $\mathbb{C}(q^2)$3 and $\mathbb{C}(q^2)$4 satisfy a piecewise Chebyshev-type recurrence:

$\mathbb{C}(q^2)$5

with initial conditions specified in terms of $\mathbb{C}(q^2)$6. These recursions are determinant (continuant) formulas for two variants of Chebyshev polynomials of the second kind, encoding an explicit combinatorial structure within the $\mathbb{C}(q^2)$7-Gaussian integers.

6. Additive Group Structure and Absence of a Deformed Multiplicative Law

The $\mathbb{C}(q^2)$8-Gaussian integers form an additive group isomorphic to $\mathbb{C}(q^2)$9, realized via the group action:

$q^2$0

This addition results from the commutativity of $q^2$1 and $q^2$2, mirroring classical $q^2$3 addition. No natural deformation of the multiplicative law of Gaussian integers is constructed. The available structure, instead, is that of a $q^2$4-deformed additive group, together with a $q^2$5-action of the Picard group $q^2$6 generated by $q^2$7, $q^2$8, $q^2$9, and the sign-flip $q$00. All classical relations hold except one, giving rise to a nontrivial central extension (Ovsienko, 2021).

7. Existence, Uniqueness, and Classical Limit

• Uniqueness: The conditions of $q$01-equivariance and $q$02 uniquely determine the map $q$03. Similarly, the operator $q$04 is uniquely determined by its commutation with $q$05 and prescribed fixed-point behavior.
• Classical Limit: As $q$06, the $q$07-deformed operators recover the classical translations:

$q$08

and

$q$09

retrieving the standard $q$10 lattice. The parameter $q$11 becomes trivial and all Chebyshev-type recursions reduce to classical real and imaginary part identities.

The construction thus produces a fully explicit, $q$12- and $q$13-equivariant deformation of the additive group of Gaussian integers, with structure deeply intertwined with $q$14-integer theory and Chebyshev polynomials, but without a currently established deformed multiplication law (Ovsienko, 2021).