A Unification of the Generalized Multiparameter Apostol-type Bernoulli, Euler, Fubini, and Genocchi Polynomials of Higher Order

Most unifications of the classical or generalized Bernoulli, Euler, and Genocchi polynomials involve unifying any two or all of the three special types of polynomials (see, [1, 4, 9, 18, 19, 21, 24–26, 30, 31]). In this paper, we introduce a new class of multiparameter Fubini-type generalized polynomials that unifies four families of higher order generalized Apostol-type polynomials such as the Apostol-Bernoulli, Apostol-Euler, Apostol-Genocchi, and Apostol-Fubini polynomials. Moreover, we obtain an explicit formula of these unified generalized polynomials in terms of the Gaussian hypergeometric function, and establish several symmetry identities. 2020 Mathematics Subject Classifications: 11B68, 11B73, 33C05, 05A10, 11B83


Introduction
In recent years, extensive researches on various families of numbers and polynomials such as the Bernoulli numbers and polynomials, Euler numbers and polynomials, Genocchi numbers and polynomials, Fubini numbers and polynomials, and also their generalizations and unifications (see, for instance the recent works of [1,3,4,10,11,17,25,26,28,31]) have become popular due to the abundance of their applications in many branches of mathematics such as in p-adic analytic number theory, umbral calculus, special functions and mathematical analysis, numerical analysis, combinatorics and other related fields. This motivates the author to obtain and explore a new unification of some of the recent generalizations of these special types of polynomials.
In this section, we present some of the known generalizations of Bernoulli, Euler, Genocchi, and Fubini polynomials of higher order. Throughout this paper, we use the usual notations N, Z, R, and C for the sets of natural numbers, integers, real numbers, and complex numbers respectively. Also, we let N 0 := N ∪ {0}, Z − := {−1, −2, −3, · · · }, and The bivariate Fubini polynomials of order α is defined through the generating function n (x, y) t n n! (see [12,13,15,17]).
In the next section, we try to unify all the previously mentioned special polynomials using a more generalized generating function.

A new class of unified generalized polynomials of higher order
Motivated by the generating relations (1), (2), and the definitions of the higher order Aposotol-type polynomials of parameters a, b, c, we consider the following unification of the generalized special types of polynomials mentioned in the previous section.

Some of the basic properties and identities for F (α)
n,k (x, y; a, b, c; λ) are given in the next theorems and corollaries.
Theorem 2. Let α and λ be arbitrary real or complex parameters. Then n,k x, y; Basic differential and integral identities of F (α) n,k (x, y; a, b, c; λ) are given in the next theorem.

Explicit formulas involving the Gaussian hypergeometric function
We now establish an explicit expression of F (r) n,k (x, y; a, b, c; λ) in terms of the Gaussian hypergeometric function 2 F 1 (a, b; c; z) which is given by Here, (q) 0 = 1, and (q) n = q(q + 1) · · · (q + n − 1) for n > 0.
Proof : Note that the left hand side of (3) can be written as and where S(j, i) is the Stirling numbers of the second kind, we obtain Using the explicit formula Using the identity n,k (x, y; a, b, c; λ) = (kr)! n kr n−kr i=0 n − kr i Finally, applying Pfaff-Kummer hypergeometric transformation Setting y = −(2 k−1 a b + 1) and λ = 2 k−1 β b 2 k−1 a b +1 in Corollary 6, we obtain an explicit formula of P

Symmetry Identities
In this section, we derive and investigate some symmetry identities for F  In [19], Lu and Srivastava defined the generalized sum of integer powers S k (n; λ) through the generating function Clearly, S k (n; 1) = S k (n).
Definition 2. Let λ be any real or complex paramete and b > 0, we define a more generalized sum of integer powers S k (n; b, λ) using the generating function Obviously, S k (n; e, λ) = S k (n; λ) and S k (n; e, 1) = S k (n). Now, we establish some symmetry identities involving these new class of unified generalized polynomials. The techniques used in here are parallel to the methods in [25,33]. Thus, we also include some results in [25] as corollaries.  Grouping factors and expanding G(t) into series, we obtain Similarly, r−l,k (vz, y; a, b, c; λ) t n n! .
Corollary 8. For u, v, m ∈ N and n ∈ N 0 , we have Setting y = −2 and k = 1, and replacing λ by λ 2 in Theorem 5, we have a symmetry identity for B Setting b = c = e and a = 1 in Theorem 5, we obtain a symmetry identity for the higher order generalized Fubini-type polynomials F Proof: Consider .
Expanding H(t) into series, we have Similarly, Comparing (14) and (15), we get the desired result.
Setting y = − 1 2 and k = 0 in Theorem 6, we obtain the following corollary.
Corollary 13. For u, v, m ∈ N and n ∈ N 0 , we have n r=0 n r Setting y = −2, k = 0 and replacing λ by λ 2 in Theorem 6, we obtain the following corollary. Setting y = − 1 2 and k = 1 in Theorem 6, we obtain the following corollary.
Corollary 15. For u, v, m ∈ N and n ∈ N 0 , we have n r=0 n r Setting a = 1 and b = c = e in Theorem 6, we obtain another symmetry identity for the higher order bivariate Fubini-type polynomials F  Taking y = −(2 k−1 a b +1) and λ = 2 k−1 β b 2 k−1 + 1 in Corollary 16, we get Theorem 3.5 of [25].
Corollary 17. For a, b > 0; β ∈ C; u, v, m ∈ N and n ∈ N 0 , we have n r=0 n r Theorem 7. For u, v, m ∈ N, n ∈ N 0 and y = −1, we have n r=0 n r  Expanding L(t) into a series, we get n,k (uz, y; a, b, c; λ) Similarly, Combining (16) and (17) gives the desired identity.
Setting y = − 1 2 and k = 0 in Theorem 7, we obtain the following corollary. Setting y = −2, k = 1 and replacing λ by λ 2 in Theorem 7, we obtain the following corollary. Setting y = − 1 2 and k = 1 in Theorem 7, we obtain the following corollary.