A $q$-Caputo Fractional Generalization of Tsallis Entropy: Series Representation and Non-Negativity Domains
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918406676021248 |
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| author | Gonzalez, Matias P. Bayron, Micolta-Riascos |
| author_facet | Gonzalez, Matias P. Bayron, Micolta-Riascos |
| contents | We introduce a fractional generalization of Tsallis entropy by acting with a $q$-Caputo operator on the generating family $\sum_i p_i^{\,x}$ evaluated at $x=1$. Concretely, we define $S_{q}^α$ through the $q$-Caputo differintegral of order $0<α<1$ and derive a closed series representation in terms of the $q$-Gamma function. The construction is anchored at the evaluation point, which ensures well-behaved limits: as $α\!\to\!1$ we recover the standard Tsallis entropy $S_q$. Finally we perform a numerical calculation to show the regions where the obtained $q$-fractional entropy $S^α_q$ can be non-negative (or negative) through the fractional parameter $α$ and the non extensive index $q$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_23254 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A $q$-Caputo Fractional Generalization of Tsallis Entropy: Series Representation and Non-Negativity Domains Gonzalez, Matias P. Bayron, Micolta-Riascos Statistical Mechanics Information Theory Mathematical Physics We introduce a fractional generalization of Tsallis entropy by acting with a $q$-Caputo operator on the generating family $\sum_i p_i^{\,x}$ evaluated at $x=1$. Concretely, we define $S_{q}^α$ through the $q$-Caputo differintegral of order $0<α<1$ and derive a closed series representation in terms of the $q$-Gamma function. The construction is anchored at the evaluation point, which ensures well-behaved limits: as $α\!\to\!1$ we recover the standard Tsallis entropy $S_q$. Finally we perform a numerical calculation to show the regions where the obtained $q$-fractional entropy $S^α_q$ can be non-negative (or negative) through the fractional parameter $α$ and the non extensive index $q$. |
| title | A $q$-Caputo Fractional Generalization of Tsallis Entropy: Series Representation and Non-Negativity Domains |
| topic | Statistical Mechanics Information Theory Mathematical Physics |
| url | https://arxiv.org/abs/2603.23254 |