{"title":"Two parameterized deformed Poisson type operator and the combinatorial moment formula","authors":"","doi":"10.1016/j.jmaa.2024.128888","DOIUrl":null,"url":null,"abstract":"<div><div>In this paper, we shall introduce two parameterized deformation of the classical Poisson random variable from the viewpoint of noncommutative probability, namely <span><math><mo>(</mo><mi>q</mi><mo>,</mo><mi>s</mi><mo>)</mo></math></span>-Poisson type operator (random variable) on the two parameterized deformed Fock space, namely, the <span><math><mo>(</mo><mi>q</mi><mo>,</mo><mi>s</mi><mo>)</mo></math></span>-Fock space constructed by the weighted <em>q</em>-deformation approach in <span><span>[11]</span></span>, <span><span>[4]</span></span> (see also <span><span>[6]</span></span>). The recurrence formula for the orthogonal polynomials of the <span><math><mo>(</mo><mi>q</mi><mo>,</mo><mi>s</mi><mo>)</mo></math></span>-deformed Poisson distribution is determined. Moreover we shall also give the combinatorial moment formula of the <span><math><mo>(</mo><mi>q</mi><mo>,</mo><mi>s</mi><mo>)</mo></math></span>-Poisson type operator by using the set partitions and the card arrangement technique with their statistics. Our method presented in this paper provides nice combinatorial interpretations to parameters, <em>q</em> and <em>s</em>. The deformation presented in this paper can be regarded as a generalization of the Al-Salam-Carlitz type, because the restricted case <span><math><mi>s</mi><mo>=</mo><mi>q</mi></math></span> recovers the <em>q</em>-Charlier polynomials of Al-Salam-Carlitz type appeared in combinatorics <span><span>[17]</span></span>.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2024-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Analysis and Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022247X24008102","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we shall introduce two parameterized deformation of the classical Poisson random variable from the viewpoint of noncommutative probability, namely -Poisson type operator (random variable) on the two parameterized deformed Fock space, namely, the -Fock space constructed by the weighted q-deformation approach in [11], [4] (see also [6]). The recurrence formula for the orthogonal polynomials of the -deformed Poisson distribution is determined. Moreover we shall also give the combinatorial moment formula of the -Poisson type operator by using the set partitions and the card arrangement technique with their statistics. Our method presented in this paper provides nice combinatorial interpretations to parameters, q and s. The deformation presented in this paper can be regarded as a generalization of the Al-Salam-Carlitz type, because the restricted case recovers the q-Charlier polynomials of Al-Salam-Carlitz type appeared in combinatorics [17].
期刊介绍:
The Journal of Mathematical Analysis and Applications presents papers that treat mathematical analysis and its numerous applications. The journal emphasizes articles devoted to the mathematical treatment of questions arising in physics, chemistry, biology, and engineering, particularly those that stress analytical aspects and novel problems and their solutions.
Papers are sought which employ one or more of the following areas of classical analysis:
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