A New Class of Generalized Polynomials Associated with Hermite and Bernoulli Polynomials

Abstract

In this thesis we consider a new class of generalized polynomials associated with Hermite and Bernoulli polynomials of degree n and order . The concepts of Bernoulli numbers, Bernoulli polynomials, generalized Bernoulli polynomials, Hermite-Bernoulli polynomials, the exponential function, present explicit formula for higher order derivatives of the generating functions of the Bernoulli polynomials and numerous properties of these polynomials and some relationships between Bernoulli numbers, Bernoulli polynomial, Generalized Bernoulli polynomials and Hermite Bernoulli polynomials can be viewed as theorems associated with the generating function of the Generalized Bernoulli polynomials. Some implicit summation formula and general symmetry identities are derived by using different analytical means and applying generating functions. These results extend some known summations and identities of generalized Bernoulli numbers and polynomials