Theorems on Genocchi Polynomials of Higher Order Arising From Genocchi Basis

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).…”

Symmetric Identities for Carlitz’s Type Higher-Order (p,q)-Genocchi Polynomials

“…A number of researchers have been looking into Bernoulli polynomials, Euler polynomials, and Genocchi polynomials (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]). This paper reviews Bernoulli polynomials, Euler polynomials, and Genocchi polynomials.…”

“…In [9], a new method to derive interesting properties related to higher-order Euler polynomials was discovered. Araci et al [2] constructed higher-order Genocchi polynomials by using the method of [9]. Higher-order Genocchi polynomials are defined as the following generating function Choi and Kim [3] discussed a nonlinear ordinary differential equation and identified higher-order Bernoulli polynomials with ordinary differential equations.…”

On (p,q)-Bernoulli, (p,q)-Euler and (p,q)-Genocchi Polynomials

“…By the same motivation, in [11], Duran and Acikgoz also introduced (p; q)-analogues of Bernoulli polynomials, Euler polynomials and Genocchi polynomials and they obtained the (p; q)-analogues of known earlier formulae. One can look at the papers [1], [2], [3], [4], [6], [13], [14], [15], [16], [17], [18], [19], [20], [21], [25], [26] for more information about the special polynomials. Now we begin with the following notations: N denotes the set of the natural numbers, N 0 denotes the set of nonnegative integers, R denotes the set of real numbers and C denotes the set of complex numbers.…”