A Note on Hermite poly-Bernoulli Numbers and Polynomials of the Second Kind

Abstract

In the paper, we introduce a new concept of poly-Bernoulli numbers and polynomials of the second kind which is called Hermite poly-Bernoulli numbers and polynomials of the second kind. We also investigate and analyse its applications in number theory, combinatorics and other fields of mathematics. The results derived here are a generalization of some known summation formulae earlier studied by Jolany et al. [17,18], Dattoli et al [14] and Pathan et al [29,30]. Keywords:Hermite polynomials, poly-Bernoulli polynomials of the second kind, Hermite poly-Bernoulli polynomials of the second kind, summation formulae, symmetric identities Cite This Article:

Hermite based Poly-Bernoulli Polynomials with a <em>q</em>-parameter

“…The Hermite based -Stirling polynomials of the second kind. We introduce the Hermite based -Stirling polynomials of the second kind is de…ned by (n; m; 0; 0) := S 2 (n; m) called the familiar Stirling numbers of the second kind (see [7], [16]).…”

“…One of the most considerable polynomials in the theory of special polynomials is the Hermite-Kampé de Fériet (or Gould-Hopper) polynomials (see [1]) and one other is Bernoulli polynomials (see [10], [16]). Nowadays, these type polynomials and their several generalizations have been studied and used by many mathematicians and physicsics, see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] and references therein. Araci et al [2] introduced a new concept of the Apostol Hermite-Genocchi polynomials by using the modi…ed Milne-Thomson's polynomials and also derived several implicit summation formulae and general symmetric identities arising from di¤erent analytical means and generating functions method.…”

“…Dattoli et al [6] applied the method of generating function to introduce new forms of Bernoulli numbers and polynomials, which were exploited to derive further classes of partial sums involving generalized many index many variable polynomials. Khan et al [7] introduce the Hermite poly-Bernoulli polynomials and numbers of the second kind and examined some of their applications in combinatorics, number theory and other …elds of mathematics. Kurt et al [8] studied on the Hermite-Kampé de Fériet based second kind Genocchi polynomials and presented some relatieonships of them.…”

On Gould–Hopper-Based Fully Degenerate Poly-Bernoulli Polynomials with a q-Parameter

“…Some of the most considerable polynomials in the theory of special polynomials are Bernoulli polynomails (see [1,2]) and the generalized Hermite-Kampé de Fériet (or Gould-Hopper) polynomials (see [3]). Recently, aforementioned polynomials and their diverse extensions have been studied and developed by lots of physicsics and mathematicians, see [1,[3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and references cited therein. Araci et al [4] considered a novel concept of the Apostol Hermite-Genocchi polynomials by using the modified Milne-Thomson's polynomials and obtained several implicit summation formulae and general symmetric identities arising from different analytical means and generating functions method.…”

“…Dattoli et al [9] applied the method of generating function to define novel forms of Bernoulli numbers and polynomials, which were exploited to get further classes of partial sums including generalized numerous index many variable polynomials. Khan et al [11,12] defined the Hermite poly-Bernoulli polynomials and numbers of the second kind and the degenerate Hermite poly-Bernoulli polynomials and numbers and analyzed many of their applications in combinatorics, number theory and other fields of mathematics. Kim et al [13][14][15] dealt with the several degenerate poly-Bernoulli polynomials and numbers.…”