Abstract
This work is a first step towards a theory of "$q$-deformed complex numbers". Assuming the invariance of the $q$-deformation under the action of the modular group I prove the existence and uniqueness of the operator of translations by~$i$ compatible with this action. Obtained in such a way $q$-deformed Gaussian integers have interesting properties and are related to the Chebyshev polynomials.
Highlights
The notion of q-deformed rational numbers was introduced in [11]
It was further extended to arbitrary real numbers in [12]
The goal of this paper is to extend the q-deformation to complex numbers
Summary
The notion of q-deformed rational numbers was introduced in [11]. It was further extended to arbitrary real numbers in [12]. Several properties of q-numbers were studied in [13, 8, 9]. The goal of this paper is to extend the q-deformation to complex numbers. We show that this can be done in a unique way. Already for the simplest case of Gaussian integers, i.e., complex numbers with integer real and imaginary parts, the obtained q-deformation has quite nontrivial properties.
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