Define the inversion operator on formal Laurent series

Develop a well-defined operator \(\tau\) on formal Laurent series that generalizes the relation between the \(q\)-deformation of a rational number and the \(q\)-deformation of its reciprocal to irrational real numbers.

Background

The paper introduces an involution τ\tau on rational functions by τ(f(q))=f(q−1)\tau(f(q))=f(q^{-1}), and uses it to construct a second quantized action of $\PGL_2(\mathbb Z)$. For rational numbers, this yields identities such as [1/x]q♯=1/[x]q−1♯[1/x]^{\sharp}_q=1/[x]^{\sharp}_{q^{-1}}.

For irrational real numbers, the qq-deformations are formal Laurent series rather than rational functions. The paper notes that the reciprocal identity cannot presently be extended because there is no clear definition of the substitution q↦q−1q\mapsto q^{-1} on the relevant space of formal Laurent series. Resolving this would extend the symmetry-based formulas from rational to irrational qq-real numbers.

References

In , it has already been noticed that for $x\in Q$, $$ \left[\frac{1}{x}\right]{\sharp} = \frac{1}{[x]_{q{-1}{\sharp} = \overline{J_q}\cdot [x]{\sharp}, $$ \noindent but this formula could not be generalized to the case of irrational numbers, as there is no clear way of defining the operator $\tau $ on formal Laurent series.

— Symmetries of the q-deformed real projective line  (2503.02122 - Jouteur, 3 Mar 2025) in Section 4.1, subsection “Focus on q-real numbers,” example following Corollary 4.4