New family of Bernoulli-type polynomials and some application

In this paper, we present a new family of generalized Bernoulli-type polynomials, as well as its numbers. In addition, we obtain some results such as algebraic and differential properties for this new family of Bernoulli-type polynomials. Likewise, the generalized Bernoulli-type polynomials matrix R-(alpha)(x) is introduced. We deduce some product formulae for R-(alpha))( x) and also, the inverse of the Bernoullitype matrix R is determined. Furthermore, we establish some explicit expressions for the Bernoullitype polynomial matrix R(x), which involve the generalized Pascal matrix and finally we study the summation formula of Euler-Maclaurin type and the Riemann zeta function applied to these Bernoullitype polynomials.