Identities Involving The Chebyshev Polynomials And Bernoulli Polynomials

Chebyshev polynomials and Bernoulli polynomials are very important in the calculation of mathematics,combinatorics,physics,and technical science.Con-sidering the close relationship with the Dirichlet L-function,Fibonacci sequence,Lucas sequence,many experts and scholars studied the properties of these two polynomials,from which they obtained a series of identities containing these two kinds of polynomials.In this paper,taking the Chebyshev polynomial and Bernoulli polynomial as the research object,we were given a new expression for the sum of these two kinds of polynomials.Firstly,we were given a new representation of the sum(?).Based on the previous re-search,by introducing a new second-order nonlinear recursive sequence C(h,j),we obtained a new expression which the right side of the equation is a linear combination of Chebyshev polynomials,fully revealed the connection between themselves.Secondly,based on the research of Professor A.Bayad and Professor D.Kim,we obtained the product sum in this form (?).As a corollary,we got some identities about Bernoulli numbers.Finally,based on the research of experts H.Walum,considered the closely relationship between Dirichlet-function and generalized Bernoulli numbers,we go further explored and proved the identities associated with them.