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Tsallis' q-analysis, new scales of interpolating spaces and q-rational functions

Daniel Alpay, Paula Cerejeiras, Uwe Kaehler

TL;DR

This work develops a Tsallis $q$-calculus framework for quantum-type reproducing-kernel spaces, introducing the $q$-Fock–Tsallis space with kernel $K_q(z,w)=e_q^{z\overline w}$ and analyzing when the associated inner product is Hilbert or Krein. It derives explicit operator identities for $M_z$, $R_0$, and $\mathbb I$ and shows how the canonical commutation relations deform with $q$, including a $q$-dependent formation of $q$-Stirling-like numbers. The paper also constructs Jordan chains in $\mathcal H(K_q)$ via generalized eigenvalue problems for $M_z^*$ and develops a Tsallis analogue of rational matrix-valued functions through the $q$-Tsallis Borel transform, establishing multiple equivalent realizations and connections to Gelfond–Leontiev-type differentiation. Collectively, these results provide a unified operator-theoretic framework that interpolates between Hardy- and Fock-like spaces in nonextensive statistics and offers tools for studying rational functions in Tsallis settings, with potential applications in quantum stochastic analysis and nonclassical signal processing.

Abstract

We are studying the fundamental tools for a quantum calculus based on the Tsallis $q$-exponential In particular we are looking at $q$-Fock spaces, structural identities, as well as rational functions in this context.

Tsallis' q-analysis, new scales of interpolating spaces and q-rational functions

TL;DR

This work develops a Tsallis -calculus framework for quantum-type reproducing-kernel spaces, introducing the -Fock–Tsallis space with kernel and analyzing when the associated inner product is Hilbert or Krein. It derives explicit operator identities for , , and and shows how the canonical commutation relations deform with , including a -dependent formation of -Stirling-like numbers. The paper also constructs Jordan chains in via generalized eigenvalue problems for and develops a Tsallis analogue of rational matrix-valued functions through the -Tsallis Borel transform, establishing multiple equivalent realizations and connections to Gelfond–Leontiev-type differentiation. Collectively, these results provide a unified operator-theoretic framework that interpolates between Hardy- and Fock-like spaces in nonextensive statistics and offers tools for studying rational functions in Tsallis settings, with potential applications in quantum stochastic analysis and nonclassical signal processing.

Abstract

We are studying the fundamental tools for a quantum calculus based on the Tsallis -exponential In particular we are looking at -Fock spaces, structural identities, as well as rational functions in this context.

Paper Structure

This paper contains 11 sections, 22 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reproducing kernel spaces
  4. Krein spaces
  5. Main operators and their properties
  6. Structural identities
  7. Commutators
  8. Some $q$-Stirling-like numbers
  9. Calculation of Jordan Chains
  10. Rational functions in the sense of Tsallis
  11. Aknowledgements

Key Result

Lemma 3.1

For $q \in (0,1)$ such that $nq \not= n-1,$ for all $n \in \mathbb{N}$, the reproducing kernel (RKHS) corresponds to a Krein space. However, when $q = \frac{n-1}{n}$ for some $n \in \mathbb{N},$ then the reproducing kernel (RKHS) has only a finite number of positive terms. In this case, the space is

Theorems & Definitions (41)

  • Lemma 3.1
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Definition 3.4
  • Theorem 3.5
  • proof
  • Lemma 3.6
  • Remark 3.7
  • Theorem 3.8
  • ...and 31 more