A Parametric Kind of Fubini Polynomials of a Complex Variable

In this paper, we propose a parametric kind of Fubini polynomials by defining the two specific generating functions. We also investigate some analytical properties (for example, summation formulae, differential formulae and relationships with other well-known polynomials and numbers) for our introduced polynomials in a systematic way. Furthermore, we consider some relationships for parametric kind of Fubini polynomials associated with Bernoulli, Euler, and Genocchi polynomials and Stirling numbers of the second kind.


Introduction
Mathematicians and other scientists have studied trigonometric functions, special numbers, and polynomials, and their applications because these functions have various mathematical usages which include derivative, integral and other algebraic properties. By using these functions with their functional equations and derivative equations, various properties of these special numbers and polynomials have been investigated (see ). By using these functions with a trigonometric function, we not only study some special families of polynomials and numbers including the Bernoulli, Euler, and Genocchi polynomials, but also derive some identities and relationships for these polynomials and numbers.
The classical Bernoulli polynomials B j (u), the classical Euler polynomials E j (u) and the classical Genocchi polynomials G j (u) are usually defined by means of the following generating functions and 2z e z + 1 respectively. Each of these polynomials has been extensively studied in many recent works, (see [18,19]).
The Geometric (also known as Fubini) polynomials [1] are defined by so that where j k are called the Stirling numbers of second kind, (see [13,17]).
On setting u = 1 in (4), we obtain where F j are called the jth Fubini numbers or ordered Bell numbers, (see [4,26]).
A few numbers of these polynomials are F 0 (u) = 1, F 1 (u) = u, F 2 (u) = u + 2u 2 , The Stirling numbers of the first kind are defined by the coefficients in the expansion of (u) j in terms of powers of u as follows, (see [14]) and the Stirling numbers of the second kind are defined by (see [15,16]) Recently, Masjed-Jamei et al. [6][7][8][9] and Srivastava et al. [23][24][25] introduced and studied the parametric kind of the two exponential generating functions e uz cos vz and e uz sin vz are defined by and where and In (2018), Kim and Ryoo [11] introduced the cosine-Bernoulli polynomials of a complex variable, the sine-Bernoulli polynomials of a complex variable and the cosine-Euler polynomials of a complex variable, the sine-Euler polynomials of a complex variable, respectively are defined as follows z j j! .
The main object of this paper is as follows. In Section 2, we consider generating a function for the parametric type of Fubini numbers and polynomials of a complex variable and give some basic properties of these polynomials. In Section 3, we derive recurrence relations, differentiation, summation formulae of parametric Fubini-type polynomials. In Section 4, we construct relationships for parametric Fubini-type polynomials associated with Bernoulli, Euler, Genocchi polynomials and Stirling numbers of the second kind.

Two Parametric Kind of the Fubini Polynomials of Complex Variable
In this section, we introduce the cosine-Fubini polynomials and sine-Fubini polynomials by splitting complex Fubini polynomials into real and imaginary parts and present some basic properties. Now, we consider the Fubini polynomials that are given by the generating function The well-known Euler's formula is defined as follows (see [11]) e (u+iv)z = e uz e ivz = e uz (cos vz + i sin vz).
Using (15) and (16), we have and From (17) and (18), we get and Definition 1. Two parametric kinds of Fubini polynomials or the cosine-Fubini polynomials F (c) j (u, v; w) and sine-Fubini polynomials F (s) j (u, v; w) for nonnegative integer j are defined by and respectively. It is clear that The first few follow immediately from this generating function:

Remark 1.
Taking u = 0 in (21) and (22), we get new type of polynomials as follows and respectively.
Proof. By (21), we have Comparing the coefficients of t n n! on both sides, we obtain (31). The proof of (32) is similar.
By using (9) and (21), we have which proves the desired result (35). The proof of (36) is similar.
Proof. Using the generating function (21), we have which gives the claimed result (43). The proof of (44) is similar.
Theorem 9. For j ≥ 0 and u 1 = u 2 . Then Proof. Equation (21) can be written as By equating the coefficients of z j j! on both sides, we get (45). The proof of (46) is similar. and Proof. Consider the following identity Using above identity by partial fraction, we find (1 + w)e uz cos vz (1 − w(e z − 1))we z = e uz cos vz 1 − w(e z − 1) + e uz cos vz we z which implies the desired result (47). The proof of (48) is similar.

Relationship between Appell-Type Polynomials
In this section, we prove some relationships for parametric Fubini-type polynomials related to Bernoulli, Euler, and Genocchi polynomials and Stirling numbers of the second kind. We start the following theorem.
and F (s) Proof. From (8) and (21), we have Replacing j by j − r in above equation, we get which gives the asserted result (55). The proof of (56) is similar.
and F (s) Proof. Replacing u by u + α in (21) and using the result ( [2], p. 250, Theorem 16), we have Replacing j by j − r in above equation, we get Equating the coefficients of z j on both sides, we get (57). The proof of (58) is similar.
and F (s) Proof. From (7) and (21), we find which provides the claimed result (59). The proof of (60) is similar.
Comparing the coefficients z j on both sides, we get (63). The proof of (64) is similar.

Conclusions
In our present investigation, we have introduced and studied systematically two parametric families of Fubini polynomials F (c) j (u, v; w) and F (s) j (u, v; w), which are defined using two specific generating functions. We have derived several fundamental properties of these parametric kinds of Fubini polynomials and such other polynomials as the parametric kind Bernoulli, Euler, and Genocchi polynomials. Lastly, we show that complex cosine-Fubini polynomials and complex sine-Fubini polynomials can be bespoke in terms of first-and second-form Stirling numbers.
Author Contributions: Conceptualization, S.K.S.; formal analysis, W.A.K.; investigation, S.K.S., W.A.K. and C.S.R.; project administration, W.A.K.; supervision, C.S.R.; funding acquisition, S.K.S; writing-original draft, S.K.S. and W.A.K. All authors contributed equally to the manuscript and all authors have read and agreed to the published version of the manuscript.