A new family of Frobenius-Genocchi polynomials and its certain properties

Department of Basic Sciences of Engineering, Faculty of Engineering and Natural Sciences, Iskenderun Tecnical University, TR-31200 Hatay, Turkey E-Mail: mtdrnugur@gmail.com Department of Mathematics, Faculty of Arts and Science, University of Gaziantep, TR-27310 Gaziantep, Turkey E-Mail: acikgoz@gantep.edu.tr Department of Economics, Faculty of Economics, Administrative and Social Science, Hasan Kalyoncu University, TR-27410 Gaziantep, Turkey E-Mail: mtsrkn@hotmail.com Corresponding Author


Introduction
The special polynomials that can be de…ned in a various ways such as by generating functions, by recurrence relations, by p-adic integrals, by the degenerate versions, etc. provide new means of analysis in the solution of a wide class of di¤erential or partial di¤erential equations often encountered in the …eld of physical problems.
Recently, using polyexponential function with unipoly function in [13] and its degenerate version in [16], several new extensions of some special polynomials such as Frobenius-Genocchi polynomials by Duran et al. [7], Bernoulli polynomials by Kim et al. [16,17], Bernoulli polynomials of the second kind by Kim et al. [19], Euler polynomials by Lee et al. [22] and Genocchi polynomials by Qin [24] have been extensively investigated.
Throughout of the paper we make use of the following notations: N := f1; 2; 3; g and N 0 = N [ f0g. Here, as usual, Z denotes the set of all integers, R denotes the set of all real numbers and C denotes the set of all complex numbers.
The Bernoulli polynomials of the second kind are de…ned by means of the following generating function When x = 0, b n (0) := b n are called the Bernoulli numbers of the second kind, cf. [16]. It is well-known from (1.7) that n (x) are the Bernoulli polynomials of order r, see [16].
The Stirling numbers of the …rst kind S 1 (n; k) and the Stirling numbers of the second kind S 2 (n; k) are de…ned (cf. [2; 4; 5; 12]) by means of the following generating functions: From (1.9), we get the following relations for n 0: Very recently, by means of the polyexponential function Ei k (t), cf. [13], de…ned by as inverse to the polylogarithm function as follows: Kim-Kim [17] performed to generalize the degenerate Bernoulli polynomials as given below: n; are called the degenerate poly-Bernoulli numbers. For k 2 Z, the type 2 degenerate poly-Euler polynomials E (k) n; (x) are de…ned, cf. [22], as follows: n; are called the type 2 degenerate poly-Euler numbers. Lee et al. [22] studied the type 2 degenerate poly-Euler polynomials and provided multifarious explicit formulas and identities.
Since Ei 1 (t) = e t 1, it is worthy to note that (1) n; (x) := B n; (x) and E (1) n; (x) := E n; (x) . In this paper, by means of the polyexponential function and degenerate exponential function, we introduce a class of new generating function for the Frobenius-Genocchi polynomials, called the type 2 degenerate poly-Frobenius-Genocchi polynomials. Then, we investigate diverse formulas and identities covering some summation formulas, derivative formula, correlations with Bernoulli polynomials and numbers, Stirling numbers of the both kinds, degenerate Frobenius-Genocchi polynomials and degenerate Frobenius-Euler polynomials. Moreover, by using the unipoly function, we consider degenerate unipoly-Frobenius-Genocchi polynomials and derive some formulas and relationships with Daehee numbers, degenerate Frobenius-Genocchi numbers and Stirling numbers of the …rst kind. Finally, an Gaussian integral representation of the Frobenius-Genocchi polynomials is derived by means of the 2-variable Hermite polynomials.

The type 2 Degenerate Poly-Frobenius-Genocchi Polynomials
Now, we consider the following De…nition 1 by means of the polyexponential function.
De…nition 1. Let k 2 Z. The type 2 degenerate poly-Frobenius-Genocchi polynomials are de…ned via the following exponential generating function (in a suitable neigbourhood of t = 0) in terms of the polyexponential function as given below: (u) is called type 2 degenerate poly-Frobenius-Genocchi numbers.
) as follows: ) as follows:  is valid for k 2 Z and n 0.
Proof. By De…nition 1, we consider that which implies the asserted result in (2.3). Now, we give the partial derivative of type 2 degenerate Frobenius-Genocchi polynomials, with the respect to x, by the following theorem. Proof. By De…nition 1, we consider that which implies the asserted result in (2.3).
A relation between the type 2 degenerate poly-Frobenius-Genocchi polynomials and the degenerate Frobenius-Genocchi polynomials is stated in the following theorem. Proof. From (1.11), we observe that Then, by (2.1), we get which means the asserted result in (2.5).
The immediate results of the Theorem 3 are stated below.  The degenerate Frobenius-Euler polynomials h n; (x; u) (cf. [14]) are de…ned as follows:  Here, we need the following lemma for Theorem 4.
Proof. From (1.11), we observe that which is the claimed result in (2.10).
Theorem 4. Let n be a nonnegative integer and k 2. Then the following identity holds: Proof. By (2.10), we consider Then, we obtain This …nalizes the proof of the theorem. Now, we give the following theorem.
Theorem 5. For k 2 Z and n 2 N 0 , we have Proof. Replacing t by e t 1 1 u in (2.1), we attain (u) t n n! .
This completes proof of the theorem. (x; u) t n n! = Ei k (log (1 + (1 u) t)) e x (t) which gives the asserted result in (2.11).

On the degenerate Unipoly-Frobenius-Genocchi Polynomials
In this section, Kim-Kim [13] introduced unipoly function u k (x jp ) attached to p being any arithmetic function that is a real or complex valued function de…ned on the set of positive integers as follows: It is readily seen that is the polylogarithm function given in (1.12). By means of the unipoly function, Kim-Kim [13] de…ned unipoly-Bernoulli polynomials as follows: 1 X n=0 B (k) n;p (x) t n n! = u k (1 e t jp ) 1 e t e xt : They provide several formulae and relations for these polynomials, see [13].