Poisson-Charlier Polynomials And Applications In Probability

It's only half a century from the day the Poisson-Charlier polynomials were raised,but its applications are extremely wide. Especially with the appearance of the q-Charlierpolynomials and the multivariate Charlier polynomials, Poisson-Charlier polynomials arebeing paid more and more attention and becoming the popular area of the mathematics.At the same time, the areas where they are applied are increasing and expending. Theynot only play an important role in mathematics and the related physics, but also be-gain to emerge in neurophysiology and nimated graphics. Besides, the Poisson-Charlierpolynomials are also an important research tool of the random matrix theory.This paper mainly provides some new properties of Poisson-Charlier polynomialson the basis of the existing literature and establishes the ralationship with Poisson ran-dom varible and Poisson Process. In the first chapter, we give the background of thePoisson-Charlier polynomials from the development of orthogonal polynomials and dis-cuss its current situation and development prospects. In the second chapter, besidesproviding definition, properties and related integral of Poisson process, we also give away called multiplicative renormalization to find the generating function, by which wecan get Poisson-Charlier polynomials directly. In the third chapter, inspired by Hermitepolynomials, we define Poisson-Charlier polynomials in derivative form, by which we givesome new properties of Poisson-Charlier polynomials in the following section. The fourthchapter is finished in two parts. In the first part, we define the Poisson-Charlier randomvarible and Poisson-Charlier function, and then discuss its mean, variance, covarianceand correlation coe?cient. In the second part, we give the martingale of Poisson processon the basis of Poisson-Charlier polynomials and its generating function and present itsintegral with respect to Poisson process.