Some Properties Of Bernoulli Polynomials

This paper is divided into three chapters.We mainly discusses some properties of Appell polynomials,such as Bernoulli polynomials,Euler polynomials,Hermite polynomials,Bell polynomials,etc,and the relationships between those polynomials.The first part introduces the research background,the history of development,the current situation of research at home and abroad and the related applications.It is designed to help readers get familiar with the origin of the Bernoulli polynomials,and understand the meaning of the Bernoulli polynomials,and understand the value of studying Bernoulli polynomials.It is also for readers to understand Bernoulli polynomials primarily,and to prepare for the second part to understand properties of Bernoulli polynomials and some related polynomial.The second part describes the properties of the Bernoulli polynomials and Euler polynomials,the Appell polynomials,and the second kind of Stirling numbers.The purpose of this section is to help readers master how to explore the properties of polynomials with the help of generation functions.Then the properties of polynomials are used for further calculations.Readers can understand why the generation functions imply relationships between polynomials and how to calculate them.The third part firstly introduces Faá di Brüno formula and Bell polynomials.Secondly,we introduce the proof of main results of the article.Finally we extend this idea to relevant polynomials and get some new results.The purpose of this section is to help readers learn to study polynomials with more complex generation functions and with more advanced techniques.Thus readers can produce concise expressions that explore relationship between this type of polynomials and other polynomials.In order to get this new purpose,we mainly use several formulas and the conclusions in the second part.Finally arrive at some new results.And on the basis of the existing results,these methods can also help us simplify the results of many results.The purpose of this thesis is to study the properties and applications of Bernoulli polynomials and some related polynomials,and to get a common method of studying polynomials.When we face some polynomials what are newly researched,there is a possible way to study their properties and further study relationships between those polynomials.