Construction of a new family of Fubini-type polynomials and its applications

This paper gives an overview of systematic and analytic approach of operational technique involves to study multi-variable special functions significant in both mathematical and applied framework and to introduce new families of special polynomials. Motivation of this paper is to construct a new class of generalized Fubini-type polynomials of the parametric kind via operational view point. The generating functions, differential equations, and other properties for these polynomials are established within the context of the monomiality principle. Using the generating functions, various interesting identities and relations related to the generalized Fubini-type polynomials are derived. Further, we obtain certain partial derivative formulas including the generalized Fubini-type polynomials. In addition, certain members belonging to the aforementioned general class of polynomials are considered. The numerical results to calculate the zeros and approximate solutions of these polynomials are given and their graphical representation are shown.


Introduction
Special polynomials and numbers have significant roles in various branches of mathematics, theoretical physics, and engineering. The problems arising in mathematical physics and engineering are framed in terms of differential equations. Most of these equations can only be treated by using various families of special polynomials which provide new means of mathematical analysis. They are widely used in computational models of scientific and engineering problems. In addition, these special polynomials allow the derivation of different useful identities in a fairly straightforward way and help in introducing new families of special polynomials. Throughout this article, we use the following notations and definitions: Let N = {1, 2, 3, . . .} and N 0 = N ∪ {0}, Z denotes the set of integer numbers, R denotes the set of real numbers and C denotes the set of complex numbers.
The Fubini-type polynomials appear in combinatorial mathematics and play an important role in the theory and applications of mathematics, thus many number theory and combinatorics experts have extensively studied their properties and obtained series of interesting results [7,12,14]. Kilar and Simsek [12] introduced the Fubini-type polynomials F (υ) n (x) of order υ, which are defined by the generating function where F (υ) n denotes the Fubini-type numbers of order υ. The Fubini-type numbers are related to Apostol-Bernoulli numbers and proven to be an effective tool in different topics in combinatorics and analysis.
The 2-variable general polynomials (2VGP) G n (x, y) [8] are defined by Over the last few years, many authors have used the operational methods combined with the monomiality principle [1] to introduce and study new families of special polynomials [9,21,[24][25][26][27]. Operational techniques are applicable to solve problems both in classical and quantum mechanics.
The 2VGP G n (x, y) are quasi-monomial [8] with respect to the following multiplicative and derivative operators: respectively. In view of the monomiality principle, the 2VGP G n (x, y) satisfy the following relations: (1.25)

Generalized Fubini-type polynomials via monomiality principle
In this section, with the help of the monomiality principle, two parametric types of the generalized Fubini-type polynomials are introduced by means of the generating functions.
Further, quasi-monomial properties and differential equations satisfied by these polynomials are established.

Theorem 1 The generating functions for the generalized Fubini-type polynomials
n (x, y, z) and G F (s,υ) n (x, y, z) are given as follows: respectively.
Proof In Eq. (1.7), replacing x and y by the multiplicative operatorM G of the 2VGP G n (x, y) and z, respectively, gives Finally, using Eq. (1.18) in the left hand side and denoting the resultant generalized Fubinitype polynomials in the right hand side by G F (c,υ) n (x, y, z), that is, we get the assertion in Eq. (2.1). Similarly, we can prove the assertion in Eq. (2.2).
In order to derive the quasi-monomial properties of Fubini-type polynomials n (x, y, z) and G F (s,υ) n (x, y, z), we prove the following results.

Theorem 2
The generalized Fubini-type polynomials G F (c,υ) n (x, y, z) and G F (s,υ) n (x, y, z) are quasi-monomial with respect to the following multiplicative and derivative operators: respectively.
Proof Differentiating Eq. (2.1) partially with respect to t gives which, in view of Eq. (2.1), becomes Now, applying the identity D x e xt ϕ(y, t) = t e xt ϕ(y, t) (2.12) in Eq. (2.11) and equating the coefficients of like powers of t in both sides of the resultant equation, we get which in view of the monomiality principle given in Eq.
Next, differentiating Eq. (2.1) partially with respect to x gives which, upon equating the coefficients of like powers of t together with the use of monomiality principle given in Eq.

Theorem 3 The generalized Fubini-type polynomials
n (x, y, z) satisfy the following differential equations: and The properties established in this section show that the operational technique provides a mechanism to obtain results for these polynomials as well as their generalizations and demonstrate the usefulness of method in problems of both physics and mathematics.

Identities and relations
In this section, by using generating functions (2.1) and (2.2), we establish various novel identities and relations including the generalized Fubini-type polynomials.

Theorem 4 The generalized Fubini-type polynomials
n (x, y, z) are defined by the following series representations: respectively.
Proof Utilizing generating relations (1.7) and (1.18) in generating relation (2.1) and making use of the Cauchy product rule in the resultant equation, we get Comparing the coefficients of the analogous powers of t in both sides of the above equation, we get the assertion in Eq. (3.1). Similarly, we can get the assertion in Eq. (3.2).

Theorem 5
The following summation formulae for the generalized Fubini-type polynomials G F (c,υ) n (x, y, z) and G F (s,υ) n (x, y, z) hold true: Proof Utilizing generating relations (1.7) and (1.19) in generating relation (2.1) and making use of the Cauchy product rule in the resultant equation, we get Comparing the coefficients of the analogous powers of t in both sides of the above equation, we get the assertion in Eq. (3.4). Similarly, we can get the assertion in Eq. (3.5).

Theorem 6
The following implicit summation formula for the generalized Fubini-type polynomials G F (c,υ) n (x, y, z) holds true: Proof In Eq. (2.1), replacing x by x + u then making use of Eq. (2.1) together with the series expansion of e ut in the resultant equation, we get which, upon replacing n by nκ in the right hand side and then comparing the coefficients of the like powers of t in both sides of the resultant equation yields the assertion in Eq. (3.7).

Theorem 7
The following implicit summation formula for the generalized Fubini-type polynomials G F (s,υ) n (x, y, z) holds true: which, upon simplification, gives the desired result.

Theorem 8
The following implicit summation formula for the generalized Fubini-type polynomials G F (c,υ) n (x, y, z) holds true: Proof Consider the following identity [11]: (3.14) Replacing t by t + s in the generating function (2.1) and making use of the identity (3.14), we have Replacing x by ω in Eq. (3.15), equating the resultant equation to Eq. (3.15) and then expanding the exponential function, we get ∞ n,κ=0 Now, making use of identity (3.14) in the right hand side of the above equation then replacing n by nl and κ by κm in the right hand side of the resultant equation gives Finally, comparing the coefficients of the like powers of t and s in both sides of Eq. (3.17) yields the assertion in Eq. (3.13).

Corollary 3 Taking z = 0 and replacing
Theorem 9 Let υ, σ ∈ N 0 , then the following identities hold true: and Proof Rewriting generating relation (2.1) in the following form: which, upon using Eqs. (1.2) and (2.1) and then after simplification yields the assertion in Eq. (3.19). The assertion in Eq. (3.20) can be proved similarly.

22)
Proof Taking z = 0 and replacing x by x + iz in Eq. (2.1), it follows that

Theorem 11
For υ ∈ N 0 , the following identities hold true: Finally, expanding the exponentials in both sides of the above equation and then after simplification, gives the assertion in Eq. (3.24). Similarly, the assertion in Eq. (3.25) can be proved.

Theorem 12
For n ∈ N 0 , the following identity holds true:

Partial derivative equations
In this section, we establish various partial derivative equations including the generalized Fubini-type polynomials G F (c,υ) n (x, y, z) and G F (s,υ) n (x, y, z) by applying partial derivative operator to generating relations (2.1) and (2.2). To achieve this, the following results are proved.

Theorem 13
Let m, n ∈ N with n m. Then the following partial derivative formulas for the generalized Fubini-type polynomials G F (c,υ) n (x, y, z) and G F (s,υ) n (x, y, z) hold true: respectively.
Proof Applying the derivative operator ∂ m ∂x m to the generating relation (2.1) gives which, in view of Eqs. (1.10) and (2.1) and after simplification, becomes n-κ-m (x, y, z) t n n! . (4.4) Finally, comparing the coefficients of t n n! on both sides of the above equation, we are led at the assertion in Eq. (4.1). The assertion (4.2) can be proved in a similar way.

Theorem 14 Let m, n ∈ N with n m. Then
Proof Replacing x by x + u and z by z + ω in Eq. (2.2) and then applying the derivative operator ∂ m ∂x m to the resultant equation, it follows that which, upon simplification, yields the desired result (4.5).

Theorem 15
Let υ, σ ∈ N 0 and m, n ∈ N with n m. Then Finally, simplifying and then equating the coefficients of the like powers of t in the resultant equation yields the assertion in Eq. (4.8). Similarly, we can prove the desired result (4.9).

Theorem 16 For n ∈ N, we have
and Remark 1 For n ∈ N and in view of Eqs. (2.1) and (2.2), the following results can be obtained: x, y, z), (4.17) and The theory of hybrid special polynomials has been one of the most emerging research topic in mathematical analysis and extensively studied to find useful properties and identities. Applications of various properties of multivariable hybrid special polynomials arise in problems of number theory, combinatorics, theoretical physics and other areas of pure and applied mathematics provide motivation for introducing a new class of generalized Fubini-type polynomials and explore its properties.
The properties and applications of these polynomials lie within the root polynomials. To explore the applications of the hybrid class of generalized Fubini-type polynomials, we have: 1. The hybrid polynomials comprising Fubini type polynomials occurs in many application not only in combinatorial analysis, but also other branches of mathematics, engineering and related areas. 2. The reason of interest for the hybrid polynomials related with truncated exponential polynomials originates from the fact that they appear in the theory of flattened Beam which assumes a role of foremost significance in optics and particularly in super-Gaussian optical resonators and plays an important role to evaluate integrals including products of special functions. 3. The motivation for the hybrid polynomials related with Laguerre polynomials is because of their intrinsic scientific significance and to the way that these polynomials are demonstrated to be natural solutions of a particular set of partial differential equations which often appear in the treatment of radiation physics problems such as the electromagnetic wave propagation and quantum beam life-time in storage rings. 4. The hybrid polynomials involving Hermite polynomials occur in probability as the Edgeworth series; in combinatorics, they arise in the umbral calculus as an example of an Appell sequence (play an important role in various problems connected with functional equations, interpolation problems, approximation theory, summation methods); in numerical analysis, they play a role in Gaussian quadrature; and in physics, they appear in quantum mechanical and optical beam transport problems.
In the next section, certain special cases of the generalized Fubini-type polynomials n (x, y, z) and G F (s,υ) n (x, y, z) are considered.

Special cases
We note that corresponding to each member belonging to the 2VGP G n (x, y), there exists a new special hybrid polynomial belonging to the generalized Fubini-type polynomials of parametric kind G F (c,υ) n (x, y, z) and G F (s,υ) n (x, y, z). The results related to these special hybrid polynomials can be obtained from the results established in the previous sections.

Gould-Hopper-Fubini-type polynomials
Since, for ϕ(y, t) = e yt r , the 2VGP G n (x, y) reduce to the Gould-Hopper polynomials H (r) n (x, y) ( Table 1(I)), for the same choices of ϕ(y, t), the generalized Fubini-type polynomials G F (c,υ) n (x, y, z) and G F (s,υ) n (x, y, z) reduce to the Gould-Hopper-Fubini-type polynomials (GHFTP) H (r) F (c,υ) n (x, y, z) and H (r) F (s,υ) n (x, y, z). Thus, by taking ϕ(y, t) = e yt r in the results established in Sects. 2-4, we can obtain the corresponding results for the GHFTP n (x, y, z), these results are listed in Tables 2 and 3. Table 2 Results for H (r) F (c,υ) n (x, y, z) κ (x, y, z) Table 3 Results for H (r) F (s,υ) n (x, y, z)

Identities and relations
n-κ (x, y, z) x n e yt r sin(zt) = 2υ and

Multiplicative and derivative operatorsM
Identities and relations

Multiplicative and derivative operatorsM
Identities and relations and

Hermite-Appell-Fubini-type polynomials
Since, for ϕ(y, t) = A(t)e yt 2 , the 2VGP G n (x, y) reduce to the Hermite-Appell polynomials H A n (x, y) ( Table 1(IV)). Therefore, for the same choice of ϕ(y, t), the generalized Fubinitype polynomials G F (c,υ) n (x, y, z) and G F (s,υ) n (x, y, z) reduce to Hermite-Appell-Fubini-type polynomials H A F (c,υ) n (x, y, z) and H A F (s,υ) n (x, y, z). Thus, by taking ϕ(y, t) = e yt r in the results established in Sects. 2-4, we can obtain the corresponding results for the Hermite-Appell-Fubini-type polynomials H A F (c,υ) n (x, y, z) and H A F (s,υ) n (x, y, z), these results are listed in Tables 8 and 9.
Stacks of zeros of GHFP H (r) F (c,v) n (x, y, z) = 0 for r = 2, v = 4, y = 3, z = 5 and 1 ≤ n ≤ 20 form a 3-D structure and are presented in Fig. 6. Mathematical problems can be investigated more effectively and more thoroughly using computers. The ability to make the figures on the computer screen empowers us to envision and produce numerous problems, analyze the properties of figures and search for new patterns.