Two-Variable Type 2 Poly-Fubini Polynomials

: In the present work, a new extension of the two-variable Fubini polynomials is introduced by means of the polyexponential function, which is called the two-variable type 2 poly-Fubini polynomials. Then, some useful relations including the Stirling numbers of the second and the ﬁrst kinds, the usual Fubini polynomials, and the higher-order Bernoulli polynomials are derived. Also, some summation formulas and an integral representation for type 2 poly-Fubini polynomials are investigated. Moreover, two-variable unipoly-Fubini polynomials are introduced utilizing the unipoly function, and diverse properties involving integral and derivative properties are attained. Furthermore, some relationships covering the two-variable unipoly-Fubini polynomials, the Stirling numbers of the second and the ﬁrst kinds, and the Daehee polynomials are acquired.


Introduction
Throughout the paper, we use N := {1, 2, 3, · · · } and N 0 = N ∪ {0}. Let C denote the set of complex numbers, R denote the set of real numbers, and Z denote the set of integers.
It is easy to see from (1) and (2) that F n x, − 1 2 = E n (x).
The polyexponential function Ei k (x) is introduced by Kim-Kim [12] as follows x n (n − 1)!n k , (k ∈ Z) (8) as inverse the polylogarithm function Li k (z) (cf. [6,[13][14][15]) given by Using the polyexponential function Ei k (x),  considered type 2 poly-Bernoulli polynomials, given by and attained several properties and formulas for these polynomials. Upon setting x = 0 in (10), β (k) n are called type 2 poly-Bernoulli numbers. We also notice that Ei 1 (z) = e z − 1. Hence, when k = 1, the type 2 poly-Bernoulli β (k) n (x) polynomials reduce to the Bernoulli polynomials B n (x) in (1). Some mathematicians have considered and examined several extensions of special polynomials via polyexponential function, cf. [5,11,13,16,17] and see also the references cited therein. For example, Duran et al. [11] defined type 2 poly-Frobenius-Genocchi polynomials by the following Maclaurin series expansion (in a suitable neighborhood of and Lee et al. [17] introduced type 2 poly-Euler polynomials given by Kim-Kim [12] also introduced unipoly function u k (x|p) attached to p being any arithmetic map which is a complex or real-valued function defined on N as follows: It is readily seen that is the ordinary polylogarithm function in (9). By utilizing the unipoly function u k (x|p),  defined unipoly-Bernoulli polynomials as follows: They derived diverse formulas and relationships for these polynomials, see [12]. The Stirling numbers of the first kind S 1 (n, k) and the second kind S 2 (n, k) are given below: From (13), for n ≥ 0, we obtain where ( From (3) and (13), we get In the following sections, we introduce a new extension of the two-variable Fubini polynomials by means of the polyexponential function, which we call two-variable type 2 poly-Fubini polynomials. Then, we derive some useful relations including the Stirling numbers of the first and the second kinds, the usual Fubini polynomials, and the Bernoulli polynomials of higher-order. Also, we investigate some summation formulas and an integral representation for type 2 poly-Fubini polynomials. Moreover, we introduce twovariable unipoly-Fubini polynomials via unipoly function and acquire diverse properties including derivative and integral properties. Furthermore, we provide some relationships covering the Stirling numbers of the first and the second kinds, the two-variable unipoly-Fubini polynomials, and the Daehee polynomials.

Two-Variable Type 2 Poly-Fubini Polynomials and Numbers
Inspired and motivated by the definition of type 2 poly-Bernoulli polynomials in (10) given by Kim-Kim [12], here, we introduce two-variable type 2 poly-Fubini polynomials by Definition 1 as follows.

Definition 1.
For k ∈ Z, we define two-variable type 2 poly-Fubini polynomials via the following exponential generating function (in a suitable neighborhood of z = 0) as given below: Upon setting x = 0 in (16), we have F (k) n (y) which we call type 2 poly-Fubini polynomials possessing the following generating function: We note that, for k = 1, the two-variable type 2 poly-Fubini polynomials reduce to the usual two-variable Fubini polynomials in (2) because of Ei 1 (z) = e z − 1. Now, we develop some relationships and formulas for two-variable type 2 poly-Fubini polynomials as follows.
holds for k ∈ Z and n ≥ 0.
A relationship involving Stirling numbers of the first kind, the two-variable Fubini polynomials, and two-variable type 2 poly-Fubini polynomials is stated by the following theorem.
Theorem 2. For k ∈ Z and n ≥ 0, we have Proof. From (13) and (17), we observe that which means the desired result (19).
Some special cases of Theorem 2 are examined below.

Corollary 1.
For k ∈ Z and n ≥ 0, we get Corollary 2. For k = 1 and n ≥ 0, we acquire The following differentiation property holds (cf. [12]) and also, the following integral representations are valid for k > 1: Theorem 3. The following relationship holds for n ∈ N 0 and k > 1. (17) and (21), for k > 1, we can write

Proof. From
holds for n ≥ 0.

Proof.
From (17), we attain and, also, we have which implies the asserted result (22).
For s ∈ C and k ∈ Z with k ≥ 1, let where Γ(s) is the classical gamma function given below: From (23), we see that η k (s) is a holomorphic map for (s) > 0, since Ei k (log(1 + z)) ≤ Ei 1 (log(1 + z)) with z ≥ 0. Thus, we have We see that the second integral in (24) converges absolutely for any s ∈ C and hence, the second term on the right hand side vanishes at non-positive integers. Therefore, we obtain Also, for (s) > 0, the first integral in (24) can be written as which defines an entire function of s. Therefore, we derive that η k (s) can be continued to an entire map of s.

Theorem 5.
For k ∈ N, the map η k (s) has an analytic continuation to a map of s ∈ C, and the special values at non-positive integers are as follows Proof. By means of (24)-(26), we acquire which is the desired relation in (27).

Now, we state a summation formula for F (k)
n (x; y) as given below.
Theorem 6. The following formula holds for k ∈ Z and n ≥ 0.
Proof. By (17), we observe that is valid for k ∈ Z and n ≥ 0.
Theorem 8. The following formula holds for k ∈ Z and n ≥ 0.
Proof. By means of (17), we acquire holds for k ∈ Z and n ≥ 0.

Theorem 10.
The following correlation hold for k ∈ Z and n ≥ 0.
Proof. Using (18), replacing z by e z − 1, we acquire that which provides the asserted result (32).

Two-Variable Unipoly-Fubini Polynomials
Using the unipoly function u k (z|p) in (11), we introduce two-variable unipoly-Fubini polynomials attached to p via the following generating function: Upon setting x = 0 in (33), we have F n,p (y) which we call unipoly-Fubini polynomials attached to p as follows We now investigate some properties of two-variable unipoly-Fubini polynomials attached to p as follows.
holds for k ∈ Z and n ≥ 1.
Proof. From (33), we observe that which means the desired result (36).
Theorem 13. The following integral representation holds for n ≥ 0 and k ∈ Z.
Taking p(n) = 1 Γ(n) in (11) gives Especially, for k = 1 in (38), we obtain which gives the following equality Lastly, we state the following theorem.