A NEW CLASS OF GENERALIZED POLYNOMIALS ASSOCIATED WITH HERMITE AND 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,v) of degree n and order alpha introduced by Derre and Simsek. The concepts of Bernoulli numbers Bn, Bernoulli polynomials B-n (x), Bernoulli numbers B-n (a; b), generalized Bernoulli polynomials B-n (x; a; b; c) of Luo et al., Hermite-Bernoulli polynomials B-H(n) (x; y) of Dattoli et al. and B-H(n)(alpha) (x; y) of Pathan are generalized to the one B-H(n)(alpha) (x; y; a; b; c) which is called the generalized polynomials depending on three positive real parameters. Numerous properties of these polynomials and some relationships between B-n, B-n (x), B-n (a; b), B-n (x; a; b; c) and B-H(n)(alpha) (x; y; a; b; c) 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 Bernoulli numbers and polynomials.