Some symmetric identities for the generalized Bernoulli, Euler and Genocchi polynomials associated with Hermite polynomials

Abstract

In 2008, Liu and Wang established various symmetric identities for Bernoulli, Euler and Genocchi polynomials. In this paper, we extend these identities in a unified and generalized form to families of Hermite–Bernoulli, Euler and Genocchi polynomials. The procedure followed is that of generating functions. Some relevant connections of the general theory developed here with the results obtained earlier by Pathan and Khan are also pointed out.

A New Family of Degenerate Poly-Genocchi Polynomials with Its Certain Properties

“…The classical Euler polynomials E n ðxÞ and the classical Genocchi polynomials G n ðxÞ are, respectively, defined by the following generating functions (see [12][13][14][15][16][17][18][19][20][21][22]):…”

A note on degenerate poly-Genocchi polynomials

“…*Genocchi polynomials are very frequently used in various problems in pure and applied mathematics related to functional equations, number theory, complex analytic number theory, Homotopy theory (stable Homotopy groups of spheres), differential topology (differential structures on spheres), theory of modular forms (Eisenstein series), -adic analytic number theory ( -adic -functions), quantum physics (quantum Groups). For instance, generating functions for Genocchi polynomials with their congruence properties, recurrence relations, computational formulae and symmetric sum involving these polynomials have been studied by many authors in recent years such as Young (2008), Araci (2014), Araci et al (2011), Açıkgöz et al (2011, Araci et al (2014aAraci et al ( , 2014b, Haroon and Khan (2018), Khan et al (2017, Khan and Haroon (2016), and Araci (2012).…”

Analytical properties of extended Hermite-Bernoulli polynomials

“…On taking α = 1, (1.8) easily reduces to (1.7). For more details about the Bernoulli numbers, Bernoulli polynomials and Hermite-Bernoulli polynomials, we refer to see, for example, [6][7][8] and the references cited therein. The aim of this article is to propose a new family of Hermite-Bernoulli polynomials in a unified and generalized form, which is given in the next section.…”