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

TL;DR

This paper introduces a two-parameter deformation of the Poisson distribution using noncommutative probability, leading to new orthogonal polynomials and a combinatorial moment formula linked to Fibonacci numbers.

Contribution

It develops a novel $(q,t)$-deformation of the Poisson operator and distribution, with a combinatorial moment formula involving set partitions and crossing/nesting statistics.

Findings

Derived a combinatorial moment formula for the $(q,t)$-Poisson operator

Established a duality between crossing and nesting statistics

Connected the structure to generalized Fibonacci numbers

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.