The Pascal Matrix Function and Its Applications to Bernoulli Numbers and Bernoulli Polynomials and Euler Numbers and Euler Polynomials

摘要

A Pascal matrix function is introduced by Call and Velleman. In this paper, we will use the function to give a unified approach in the study of Bernoulli numbers and Bernoulli polynomials. Many well-known and new properties of the Bernoulli numbers and polynomials can be established by using the Pascal matrix function. The approach is also applied to the study of Euler numbers and Euler polynomials. Finally, an application of an extension of Bernoulli polynomials to the numerical integrations is given.