q-Fock Space of q-Analytic Functions and its realization in \(L^2(\mathbb C^2; e^{-|z|^2}dxdy)\)

We introduce a q-deformation of the Fock space of holomorphic functions on \(\mathbb C\), based on a geometric definition of q-analyticity. This definition is inspired by a standard construction in complex differential geometry. Within this framework, we define q-analytic monomials \(z^n_q\) and construct the associated q-Fock space as a Hilbert space with orthonormal basis \(\{z^n_q/\sqrt{[n]_q!}\}_{n\geq 0}\). The reproducing kernel of this space is computed explicitly, and q-position and q-momentum operators are introduced, satisfying q-deformed commutation relations. We show that the q-monomials \(z^n_q\) can be expanded in terms of complex Hermite polynomials, thereby providing a realization of the q-Fock space as a subspace of \(L^2(\mathbb C^2; e^{-|z|^2}dxdy)\). Finally, we define a q-Bargmann transform that maps suitable q-Hermite functions into our q-Fock space and acts as a unitary isomorphism. Our construction offers a geometric and analytic approach to q-function theory, complementing recent operator-theoretic models.