Bernoulli Polynomials And Euler Polynomials Based On The Akiyama-Tanigawa Algorithm

Bernoulli polynomials,Bernoulli polynomials of higher order,Euler polynomials and Euler polynomials of higher order have a wide range of applications in analytic number theory and function theory.The Akiyama-Tanigawa algorithm is the algorithm for computing Bernoulli numbers.Using the Akiyama-Tanigawa algorithm,we obtain a kind of closed formulas for Bernoulli polynomials of higher order and Euler polynomials of higher order related to the weighted Stirling numbers of the second kind.As corollaries,two combinatorial identities related to the Stirling numbers of two kinds follow.The main content of this thesis can be summarized as follows:1.Study some of concepts and theorems of Bernoulli polynomials,Euler polynomials and Stirling numbers.Then investigative the Akiyama-Tanigawa algorithm.2.Using the Akiyama-Tanigawa algorithm,we obtain the computing formulas for Bernoulli polynomials and Euler polynomials related to the weighted Stirling numbers of the second kind.As a corollary,an combinatorial identity contacting Bernoulli numbers and the Stirling numbers of the second kind follow.3.Using the Akiyama-Tanigawa algorithm,we acquire the computing formulas for Bernoulli polynomials of higher order and Euler polynomials of higher order related to the weighted Stirling numbers of the second kind.As a corollary,an combinatorial identity related to the Stirling numbers of two kinds follow.