{"title":"Probabilistic Bernoulli and Euler Polynomials","authors":"T. Kim, D. S. Kim","doi":"10.1134/s106192084010072","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> Let <span>\\(Y\\)</span> be a random variable whose moment generating function exists in a neighborhood of the origin. The aim of this paper is to introduce and study the probabilistic extension of Bernoulli polynomials and Euler polynomials, namely the probabilistic Bernoulli polynomials associated <span>\\(Y\\)</span> and the probabilistic Euler polynomials associated with <span>\\(Y\\)</span>. Also, we introduce the probabilistic <span>\\(r\\)</span>-Stirling numbers of the second associated <span>\\(Y\\)</span>, the probabilistic two variable Fubini polynomials associated <span>\\(Y\\)</span>, and the probabilistic poly-Bernoulli polynomials associated with <span>\\(Y\\)</span>. We obtain some properties, explicit expressions, certain identities and recurrence relations for those polynomials. As special cases of <span>\\(Y\\)</span>, we treat the gamma random variable with parameters <span>\\(\\alpha,\\beta > 0\\)</span>, the Poisson random variable with parameter <span>\\(\\alpha >0\\)</span>, and the Bernoulli random variable with probability of success <span>\\(p\\)</span>. </p><p> <b> DOI</b> 10.1134/S106192084010072 </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.7000,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Russian Journal of Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1134/s106192084010072","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
Abstract
Let \(Y\) be a random variable whose moment generating function exists in a neighborhood of the origin. The aim of this paper is to introduce and study the probabilistic extension of Bernoulli polynomials and Euler polynomials, namely the probabilistic Bernoulli polynomials associated \(Y\) and the probabilistic Euler polynomials associated with \(Y\). Also, we introduce the probabilistic \(r\)-Stirling numbers of the second associated \(Y\), the probabilistic two variable Fubini polynomials associated \(Y\), and the probabilistic poly-Bernoulli polynomials associated with \(Y\). We obtain some properties, explicit expressions, certain identities and recurrence relations for those polynomials. As special cases of \(Y\), we treat the gamma random variable with parameters \(\alpha,\beta > 0\), the Poisson random variable with parameter \(\alpha >0\), and the Bernoulli random variable with probability of success \(p\).
期刊介绍：
Russian Journal of Mathematical Physics is a peer-reviewed periodical that deals with the full range of topics subsumed by that discipline, which lies at the foundation of much of contemporary science. Thus, in addition to mathematical physics per se, the journal coverage includes, but is not limited to, functional analysis, linear and nonlinear partial differential equations, algebras, quantization, quantum field theory, modern differential and algebraic geometry and topology, representations of Lie groups, calculus of variations, asymptotic methods, random process theory, dynamical systems, and control theory.
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