Q-analysis: A Framework for q-Deformations

• Q-analysis is a combinatorial and functional framework that uses a deformation parameter q to generalize quantum calculus and extend classical analytic structures.
• It introduces q-deformed spaces like the q-Fock-Tsallis space, establishing Hilbert or Krein spaces via reproducing kernels and explicit expansion coefficients.
• New operator identities and constructs such as q-rational functions interpolate between classical Fock and Hardy spaces, linking nonextensive statistical mechanics with advanced operator theory.

Q-analysis is a combinatorial and functional framework for decomposing and analyzing complex structures using parameterized algebraic and analytical tools, notably in quantum calculus, operator theory, and mathematical physics. Recent advances introduce generalized function spaces, structural identities, and rationality concepts indexed by a deformation parameter $q$, motivated by Tsallis nonextensive statistical mechanics and elucidating connections between distinct areas such as Fock and Hardy spaces, non-Hilbertian structures, and rational transforms.

1. Quantum Calculus with the Tsallis $q$-Exponential

Q-analysis, as developed in the referenced work, centers on a quantum calculus utilizing the Tsallis $q$-exponential

$$e_q(x) := \begin{cases} [1 + (1-q)x]^{\frac{1}{1-q}}, & 1 + (1-q)x \geq 0 \ 0, & \text{otherwise} \end{cases}$$

and logarithm

$\ln_q x := \frac{x^{1-q} - 1}{1-q}, \quad x > 0.$

As $q$ varies, these interpolating functions recover standard analytic structures: $e_1(x) = e^x$ (standard exponential), $e_2(x) = \frac{1}{1-x}$ (Hardy space kernel). This formalism replaces the classical exponential underpinning Hida’s white noise theory and Boltzmann-Gibbs statistical mechanics, thereby generalizing analytic methods to a parameterized family indexed by $q$, relevant to nonextensive thermostatistics and quantum probability.

2. Construction and Properties of $q$-Fock-Tsallis Spaces

Central is the introduction of the $q$-Fock-Tsallis space $F_q$, a reproducing kernel Hilbert or Krein space defined by the kernel

$$K_q(z, w) = e_q(z\overline{w}) = [1 + (1-q)z\overline{w}]^{\frac{1}{1-q}}.$$

For $q=1$, $K_1(z, w) = e^{z\overline{w}}$ (classical Fock space); for $q=2$, $K_2(z,w) = (1-z\overline{w})^{-1}$ (Hardy space). The series expansion

$$K_q(z, w) = \sum_{k=0}^\infty \frac{(1-q)^k (\frac{1}{1-q})_k}{k!} (z\overline{w})^k$$

invokes the lowering factorial $(a)_k$. The coefficients $\alpha_k$ define whether the space is Hilbert (for $q \geq 1$, all $\alpha_k > 0$) or Krein (for $q \in (0,1)$, sign alternation past some threshold $k_0$). Finite-dimensional cases arise at special parameter values $q = \frac{k-1}{k}$. The inner product structure is

$$\| f \|^2_{F_q} = \sum_{k=0}^\infty |f_k|^2 |\gamma_k|,\quad \gamma_k = \frac{k!}{\alpha_k}.$$

This generalizes classical reproducing kernel spaces and links them via a $q$-parameter scale.

3. New Operator Identities and Scale Interpolation

A key structural innovation is the identification of operator relations interpolating between Fock and Hardy spaces: $(q-1)M_z^* = R_0 + (q-2)\mathbb{I}^*$ where $M_z$ is multiplication, $R_0$ is the backward shift, and $\mathbb{I}$ is the integration operator. For $q=1$, this reduces to Fock space relations; for $q=2$, to Hardy space relations. New commutator formulas, e.g.,

$$[M_z, M_z^*] z^k = -1 \quad (k=0),\quad [R_0, R_0^*]z^k = (q-2)(R_0 \mathbb{I}^2 R_0) z^k,$$

along with the introduction of $q$-Stirling-like numbers, generalize the classical combinatorial framework, yielding a continuous scale of spaces depending on $q$ and supporting analysis of both indefinite (Krein) and definite (Hilbert) regimes.

4. $q$-Rational Functions and the $q$-Tsallis Borel Transform

The notion of $q$-rationality is established: a function $f$ is $q$-rational if the span $\{f, M_z^* f, (M_z^*)^2 f, \ldots\}$ is finite-dimensional, which is equivalent to being the image under a $q$-Tsallis Borel transform,

$$B_q F(z) = \sum_{k=0}^\infty \frac{(1-q)^k (\frac{1}{1-q})_k}{k!} F_k z^k$$

for a classical rational function $F$. Several equivalent realizations are given:

• The existence of $F(z) = C(e_q(zA))B$ for suitable matrices,
• Finite-rank Hankel matrices with explicit $q$-deformed blocks,
• Series representation matching the expansion coefficients.

This framework connects classical rational function theory, functional analysis, and operator theory in a $q$-indexed formalism, extending the reach of classical Borel transforms.

5. Generalization via Gelfond-Leontiev Operators

The theory is broadened further by replacing the Tsallis exponent $e_q$ with a general entire function $\varphi$ and introducing Gelfond-Leontiev differentiation. This enables development of analogous reproducing kernel spaces, operator identities, and rationality concepts for even more general settings, suggesting a deep structural connection between disparate areas including operator theory, quantum calculus, and mathematical physics.

6. Summary of Advances and Directions

Q-analysis, as formalized in this framework, provides:

Feature	Classical Case	$q$-Analysis (This Work)
Kernel function	$e^{z\overline{w}}$, $(1-z\overline{w})^{-1}$	$[1 + (1-q)z\overline{w}]^{1/(1-q)}$
Space structure	Fock/Hardy (Hilbert) Spaces	Hilbert or Krein, depending on $q$
Operator relations	Fixed (Fock/Hardy)	$q$-parametric, interpolating identities
Rational functions	Classical, via Hankel matrices	$q$-rational, Tsallis Borel transform
Combinatorial objects	Stirling numbers	$q$-Stirling analogs
Generalization	Limited	Arbitrary entire functions, Gelfond-Leontiev

These developments establish a unified framework for analyzing spaces, operators, and functions across a spectrum of analytic regimes, controlled by deformation parameters such as $q$. The explicit structural formulas, operator identities, rational function constructions, and expansion coefficients provide direct tools for further research in quantum probability, operator theory, nonextensive physics, and functional analysis. Future directions include the exploration of spectral properties, connections to quantum information, and applications to complex dynamical systems and quantum stochastic processes.