A NEW CLASS OF GENERALIZED POLYNOMIALS ASSOCIATED WITH HERMITE AND POLY-BERNOULLI POLYNOMIALS

In this paper, we introduce a new class of generalized polynomials associated with the modified Milne-Thomson's polynomials Phi((alpha))(n) (x, nu) of degree n and order alpha introduced by Dere and Simsek. The concepts of poly-Bernoulli numbers, poly-Bernoulli polynomials, Hermite-Bernoulli polynomials and generalized Hermite-Bernoulli polynomials are generalized to polynomials of three positive real parameters. Numerous properties of these polynomials and some relations are established. Some implicit summation formulae 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 poly-Bernoulli numbers and polynomials.