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
