On Type 2 Degenerate Poly-Frobenius-Genocchi Polynomials and Numbers

In this paper, we consider a class of new generating function for the Frobenius-Genocchi polynomials, called the type 2 degenerate poly-Frobenius-Genocchi polynomials, by means of the polyexponential function. Then, we investigate diverse explicit expressions and some identities for those polynomials.


Introduction
Special polynomials have their origin in the solution of the di¤erential equations (or partial di¤erential equations) under some conditions. Special polynomials can be de…ned in a various ways such as by generating functions, by recurrence relations, by p-adic integrals in the sense of fermionic and bosonic, by degenerate versions, etc.
Kim-Kim have introduced polyexponential function in [18] and its degenerate version in [20], [21]. By making use of aforementioned function, they have introduced a new class of some special polynomials. This idea provides a powerfool tool in order to de…ne special numbers and polynomials by making use of polyexponential function. One may see that the notion of polyexponential function form a special class of polynomials because of their great applicability, cf. [12, 18-22, 26, 27, 29, 31]. The importance of these polynomials would be to …nd applications in analytic number theory, applications in classical analysis and statistics, cf. .
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 integers, R denotes the set of real numbers and C denotes the set of complex numbers.
Khan and Srivastava [17] introduced a new class of the generalized Apostol type Frobenius-Genocchi polynomials and investigated some properties and relations including implicit summation formulae and various symmetric identities. Moreover a relation in between Array-type polynomials, Apostol-Bernoulli polynomials and generalized Apostol-type Frobenius-Genocchi polynomials is also given in [17]. Wani et al. [33] considered Gould-Hopper based Frobenius-Genocchi polynomials and then, summation formulae and operational rule for these polynomials.
For k 2 Z, the type 2 degenerate poly-Euler polynomials E (k) n; (x) are de…ned, cf. [29], as follows: n; are called the type 2 degenerate poly-Euler numbers. Lee et al. [29] 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

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) including the polyexponential function as given below: At the value x = 0 in (2.1), G (F;k) n; (0; u) := G (F;k) n; (u) will be called type 2 degenerate poly-Frobenius-Genocchi numbers. is valid for k 2 Z and n 0.
Proof. By De…nition 1, we consider that (u) t n n!
is valid for k 2 Z and n 0.
Proof. By De…nition 1, we consider that which implies the asserted result in (2.2).
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 (1.15), we observe that Then, by (2.1), we get  Here, we give the following lemma.
Lemma 1. For k 2 Z and n 0, we have Proof. From (1.15), we observe that which is the claimed result in (2.7). ; m k 1 (1 u) Proof. By (2.7), we consider Then, we obtain (u) t n n! .