MMN-1684

# A new class of generalized polynomials associated with Hermite and poly-Bernoulli polynomials

• M. A. Pathan, Centre for Mathematical and Statistical Sciences (CMSS), KFRI, Peechi P.O., Thrissur, Kerala-680653, India, mapathan@gmail.com
• Waseem A. Khan, Department of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, P.O Box 1664, Al Khobar 31952, Saudi Arabia, wkhan1@pmu.edu.sa

## Abstract

In this paper, we introduce a new class of generalized polynomials associated with the modified Milne-Thomson's polynomials $\Phi_{n}^{(\alpha)}(x,\nu)$ of degree n and order $\alpha$ introduced by Dere and Simsek. The concepts of Poly-Bernoulli numbers $B_n^{(k)}(a,b)$, Poly-Bernoulli polynomials $B_n^{(k)}(x,a,b)$ of Joalny et al, Hermite-Bernoulli polynomials ${_HB}_n(x,y)$ of Dattoli et al and ${_HB}_n^{(\alpha)} (x,y)$ of Pathan et al are generalized to the one ${_HB}_n^{(k)}(x,y,a,b,c)$ which is called the generalized polynomials depending on three positive real parameters. Numerous properties of these polynomials and some relationships between $B_n^{(k)}(a,b)$, $B_n^{(k)}(x,a,b)$, $B_n^{(k)}(x;a,b,c)$ and ${}_HB_n^{(k)}(x,y;a,b,c)$ 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 Poly-Bernoulli numbers and polynomials.

Vol. 22 (2021), No. 1, pp. 317-330
DOI: 10.18514/MMN.2021.1684