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arXiv · 2508.12659

A Poisson Type Operator Deformed by Generalized Fibonacci Numbers and Its Combinatorial Moment Formula

Nobuhiro Asai·Marek Bożejko·Lahcen Oussi·Hiroaki Yoshida
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Abstract

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.

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Nobuhiro Asai, Marek Bożejko, Lahcen Oussi, Hiroaki Yoshida. 2025-08-18. A Poisson Type Operator Deformed by Generalized Fibonacci Numbers and Its Combinatorial Moment Formula. https://arxiv.org/abs/2508.12659

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