A Poisson Type Operator Deformed by Generalized Fibonacci Numbers and Its Combinatorial Moment Formula
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2025
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| author | Asai, Nobuhiro Bożejko, Marek Oussi, Lahcen Yoshida, Hiroaki |
| author_facet | Asai, Nobuhiro Bożejko, Marek Oussi, Lahcen Yoshida, Hiroaki |
| contents | We introduce a two-parameter deformation of the classical Poisson distribution from the viewpoint of noncommutative probability theory, by defining a $(q,t)$-Poisson type operator (random variable) on the $(q,t)$-Fock space \cite{Bl12} (See also \cite{BY06, AY20}). From the analogous viewpoint of the classical Poisson limit theorem in probability theory, we are naturally led to a family of orthogonal polynomials, which we call the $(q,t)$-Charlier polynomials. These generalize the $q$-Charlier polynomials of Saitoh-Yoshida \cite{SY00a, SY00b} and reflect deeper combinatorial symmetries through the additional deformation parameter $t$. A central feature of this paper is the derivation of a combinatorial moment formula of the $(q,t)$-Poisson type operator and the $(q,t)$-Poisson distribution. This is accomplished by means of a card arrangement technique, which encodes set partitions together with crossing and nesting statistics. The resulting expression naturally exhibits a duality between these statistics, arising from a structure rooted in generalized Fibonacci numbers. Our approach provides a concrete framework where methods in combinatorics and theory of orthogonal polynomials are used to investigate the probabilistic properties arising from the $(q,t)$-deformation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_12659 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Poisson Type Operator Deformed by Generalized Fibonacci Numbers and Its Combinatorial Moment Formula Asai, Nobuhiro Bożejko, Marek Oussi, Lahcen Yoshida, Hiroaki Combinatorics Mathematical Physics Probability 46L53, 05A17, 33D45, 60E99, 05A30 We introduce a two-parameter deformation of the classical Poisson distribution from the viewpoint of noncommutative probability theory, by defining a $(q,t)$-Poisson type operator (random variable) on the $(q,t)$-Fock space \cite{Bl12} (See also \cite{BY06, AY20}). From the analogous viewpoint of the classical Poisson limit theorem in probability theory, we are naturally led to a family of orthogonal polynomials, which we call the $(q,t)$-Charlier polynomials. These generalize the $q$-Charlier polynomials of Saitoh-Yoshida \cite{SY00a, SY00b} and reflect deeper combinatorial symmetries through the additional deformation parameter $t$. A central feature of this paper is the derivation of a combinatorial moment formula of the $(q,t)$-Poisson type operator and the $(q,t)$-Poisson distribution. This is accomplished by means of a card arrangement technique, which encodes set partitions together with crossing and nesting statistics. The resulting expression naturally exhibits a duality between these statistics, arising from a structure rooted in generalized Fibonacci numbers. Our approach provides a concrete framework where methods in combinatorics and theory of orthogonal polynomials are used to investigate the probabilistic properties arising from the $(q,t)$-deformation. |
| title | A Poisson Type Operator Deformed by Generalized Fibonacci Numbers and Its Combinatorial Moment Formula |
| topic | Combinatorics Mathematical Physics Probability 46L53, 05A17, 33D45, 60E99, 05A30 |
| url | https://arxiv.org/abs/2508.12659 |