Some New Classes of Generalized Hermite-Based Apostol-Euler and Apostol-Genocchi Polynomials

Abstract

In this paper, we introduce a new class of generalized Apostol-Hermite-Euler polynomials and Apostol-Hermite-Genocchi polynomials and derive some implicit summation formulae by applying the generating functions. These results extend some known summations and identities of generalized Hermite-Euler polynomials studied by Dattoli et al, Kurt and Pathan.

Multiple-Poly-Bernoulli Polynomials of the Second Kind Associated with Hermite Polynomials

“…These results extend some known summations and identities of generalized Hermite multipoly-Bernoulli numbers and polynomials of the second kind studied by Qi et al [11], Dattoli et al [5], Khan [7] and Pathan and Khan [9,10].…”

“…n (x) are called the higher order Bernoulli polynomials of the first kind given by the generating function (see [7,8,9,10]):…”

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

“…Replacing n by n-p in the above equation and comparing the coefficients of t n , we get the result (2.11). x y the same considerations as developed for the ordinary Hermite and related polynomials in Khan et al [21] and Hermite-Bernoulli polynomials in Pathan and Khan [29][30][31][32][33][34]…”

“…and the Stirling number of the second kind is defined by generating function to be ( ) Recently many mathematicians have studied the symmetric identities on some special polynomials see for details [29][30][31][32][33][34]43,44]. Some of mathematicians also investigated some applications of poly-Bernoulli numbers and polynomials of the second kind cf.…”

Degenerate Hermite poly-Bernoulli numbers and polynomials with q-parameter

“…Problems of this type arise, for example, in the computation of the higher-order moments of a distribution or to evaluate transition matrix elements in quantum mechanics. In [7], [8], [9], [19], [20], [21], [22], it has been shown that the summation formulae of special functions, encountered in applications ranging from electromagnetic process to combinatorics can be written in terms of Hermite polynomials of more than one variable.…”