A new class of partially degenerate Hermite-Genocchi polynomials

Abstract

In this paper, firstly we introduce not only partially degenerate Hermite-Genocchi polynomials, but also a new generalization of degenerate Hermite-Genocchi polynomials. Secondly, we investigate some behaviors of these polynomials. Furthermore, we establish some implicit summation formulae and symmetry identities by making use of the generating function of partially degenerate Hermite-Genocchi polynomials. Finally, some results obtained here extend well-known summations and identities which we stated in the paper.

On the generalized Apostol-type Frobenius-Genocchi polynomials

“…In this section, we establish symmetric identities for the generalized Apostol type Frobenius-Genocchi polynomials by applying the generating function (2.1). The results extends some known identities of Khan et al [3][4][5] and Pathan and Khan [8][9][10]. Proof.…”

“…Let α ∈ Z, λ ∈ C, a, b, c ∈ R + , a b and x ∈ R. The generalized Apostol-Bernoulli, Euler and Genocchi polynomials with the parameters are given by means of the following generating function as follows (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]): Recently, Kurt et al [1] and Simsek [11,12] introduced the Apostol type Frobenius-Euler polynomials as follows.…”

Certain Results on (p,q)-Hermite Based Apostol Type Frobenius-Euler Polynomials

“…We now begin with the following theorem. ; (see [11][12][13][14][15] Finally on equating the coe¢ cients of the like powers of t and w in the above equation, we get the required result.…”

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).…”