Abstract
Recently, Kim-Kim (2019) introduced polyexponential and unipoly functions. By using these functions, they defined type 2 poly-Bernoulli and type 2 unipoly-Bernoulli polynomials and obtained some interesting properties of them. Motivated by the latter, in this paper, we construct the poly-Genocchi polynomials and derive various properties of them. Furthermore, we define unipoly Genocchi polynomials attached to an arithmetic function and investigate some identities of them.
Highlights
The study of the generalized versions of Bernoulli and Euler polynomials and numbers was carried out in [1,2]
We mention that the study of a generalized version of the special polynomials and numbers can be done for the transcendental functions like hypergeometric ones
The poly-Bernoulli numbers are defined by means of the polylogarithm functions and represent the usual Bernoulli numbers when k = 1
Summary
The study of the generalized versions of Bernoulli and Euler polynomials and numbers was carried out in [1,2]. Various special polynomials and numbers regained the interest of mathematicians and quite a few results have been discovered. Kim-Kim introduced polyexponential and unipoly functions [9] By using these functions, they defined type 2 poly-Bernoulli and type 2 unipoly-Bernoulli polynomials and obtained several interesting properties of them. We consider poly-Genocchi polynomials which are derived from polyexponential functions. We give explicit expressions and identities involving those polynomials It is well known, the Bernoulli polynomials of order α are defined by their generating function as follows (see [1,2,3,17,18]): α.
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