Deformations and q-convolutions. Old and new results
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916125729619968 |
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| author | Bozejko, Marek Bozejko, Wojciech |
| author_facet | Bozejko, Marek Bozejko, Wojciech |
| contents | This paper is the survey of some of our results related to $q$-deformations of the Fock spaces and related to $q$-convolutions for probability measures on the real line $\mathbb{R}$. The main idea is done by the combinatorics of moments of the measures and related $q$-cumulants of different types. The main and interesting $q$-convolutions are related to classical continuous (discrete) $q$-Hermite polynomial. Among them are classical ($q=1$) convolutions, the case $q=0$, gives the free and Boolean relations, and the new class of $q$-analogue of classical convolutions done by Carnovole, Koornwinder, Biane, Anshelovich, and Kula. The paper contains many questions and problems related to the positivity of that class of $q$-convolutions. The main result is the construction of Brownian motion related to $q$-Discrete Hermite polynomial of type I. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_06405 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Deformations and q-convolutions. Old and new results Bozejko, Marek Bozejko, Wojciech Mathematical Physics Combinatorics Probability This paper is the survey of some of our results related to $q$-deformations of the Fock spaces and related to $q$-convolutions for probability measures on the real line $\mathbb{R}$. The main idea is done by the combinatorics of moments of the measures and related $q$-cumulants of different types. The main and interesting $q$-convolutions are related to classical continuous (discrete) $q$-Hermite polynomial. Among them are classical ($q=1$) convolutions, the case $q=0$, gives the free and Boolean relations, and the new class of $q$-analogue of classical convolutions done by Carnovole, Koornwinder, Biane, Anshelovich, and Kula. The paper contains many questions and problems related to the positivity of that class of $q$-convolutions. The main result is the construction of Brownian motion related to $q$-Discrete Hermite polynomial of type I. |
| title | Deformations and q-convolutions. Old and new results |
| topic | Mathematical Physics Combinatorics Probability |
| url | https://arxiv.org/abs/2311.06405 |