A new class of generalized polynomials associated with Hermite and Bernoulli polynomials

  • M. A. Pathan Centre for Mathematical Sciences, Pala, 686574, Kerala, India
  • Waseem A. Khan Department of Mathematics, Integral University, Lucknow-226026, (India)}
Keywords: Hermite polynomials, Bernoulli polynomials, Hermite-Bernoulli polynomials, summation formulae, symmetric identities

Abstract

In this paper, we introduce a new class of generalized  polynomials associated with  the modified Milne-Thomson's polynomials Φ_{n}^{(α)}(x,ν) of degree n and order α introduced by  Derre and Simsek.The concepts of Bernoulli numbers B_n, Bernoulli polynomials  B_n(x), generalized Bernoulli numbers B_n(a,b), generalized Bernoulli polynomials  B_n(x;a,b,c) of Luo et al, Hermite-Bernoulli polynomials  {_HB}_n(x,y) of Dattoli et al and {_HB}_n^{(α)} (x,y) of Pathan  are generalized to the one   {_HB}_n^{(α)}(x,y,a,b,c) which is called  the generalized  polynomial 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 {}_HB_n^{(α)}(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

Author Biographies

M. A. Pathan, Centre for Mathematical Sciences, Pala, 686574, Kerala, India

Scientist

Centre for Mathamatical Sciences,Pala,686574,Kerala,India

Waseem A. Khan, Department of Mathematics, Integral University, Lucknow-226026, (India)}

Assistant Professor

Department of Mathematics, Integral University,Lucknow-226026, (India)}

Published
2015-05-04
Section
Articoli