NEW GENERALIZED APOSTOL-FROBENIUS-EULER POLYNOMIALS AND THEIR MATRIX APPROACH

In this paper, we introduce a new extension of the generalized ApostolFrobenius-Euler polynomials H n (x; c, a;λ;u). We give some algebraic and differential properties, as well as, relationships between this polynomials class with other polynomials and numbers. We also, introduce the generalized ApostolFrobenius-Euler polynomials matrix U[m−1,α](x; c, a;λ;u) and the new generalized Apostol-Frobenius-Euler matrix U[m−1,α](c, a;λ;u), we deduce a product formula for U[m−1,α](x; c, a;λ;u) and provide some factorizations of the Apostol-Frobenius-Euler polynomial matrix U[m−1,α](x; c, a;λ;u), which involving the generalized Pascal matrix.


Introduction
It is well-known that generalized Frobenius-Euler polynomial H (α) n (x; u) of order α is defined by means of the following generating function where u ∈ C and α ∈ Z. Observe that H (1) n (x; u) = H n (x; u) denotes the classical Frobenius-Euler polynomials and H (α) n (0; u) = H (α) n (u) denotes the Frobenius-Euler numbers of order α. H n (x; −1) = E n (x) denotes the Euler polynomials (see [2,7]).
In the present paper, we introduce a new class of Frobenius-Euler polynomials considering the work of [8], we give relationships between this polynomials whit other polynomials and numbers, as well as the generalized Apostol-Frobenius-euler polynomials matrix.
The paper is organized as follows. Section 2 contains the definitions of Apostoltype Frobenius-Euler and generalized Apostol-Frobenius-Euler polynomials and some auxiliary results. In Section 3, we define the generalized Apostol-type Frobenius-Euler polynomials and prove some algebraic and differential properties of them, as well as their relation with the Stirling numbers of second kind. Finally, in Section 4 we introduce the generalized Apostol-type Frobenius-Euler polynomial matrix, derive a product formula for it and give some factorizations for such a matrix, which involve summation matrices and the generalized Pascal matrix of first kind in base c, respectively.

Previous Definitions and Notations
Throughout this paper, we use the following standard notions: N = {1, 2, . . .}, N 0 = {0, 1, 2, . . .}, Z denotes the set of integers, R denotes the set of real numbers and C denotes the set of complex numbers. Furthermore, (λ 0 ) = 1 and where k ∈ N, λ ∈ C. For the complex logarithm, we consider the principal branch. All matrices are in M n+1 (K), the set of all (n + 1) × (n + 1) matrices over the field K, with K = R or C. Also, for i, j any nonnegative integers we adopt the following convention i j = 0, whenever j > i.
Now, let us givel some properties of the generalized Apostol-type Frobenius-Euler polynomials and generalized Apostol-type Frobenius-Euler polynomials with parameters λ, a, c, order α (see [4,8,11]).  The Jacobi polynomials of the degree n y orde (α, β), with α, β > −1, the n-th Jacobi polynomial P (α,β) n (x) may be defined through Rodrigues' formula The relationship between the n-th monomial x n and the n-th Jacobi polynomial P (α,β) n (x) may be written as x n = 1 2(n + 1) Definition 2.2. Let x be any nonzero real number. For c ∈ R + , the generalized Pascal matrix of first kind in base c P c [x] is an (n + 1) × (n + 1) matrix whose entries are given by (see [13,14]) An immediate consequence of the remarks above is the following proposition.

Proposition 2.3 (Addition Theorem of the argument). For x, y ∈ R is fulfilled
be the generalized Pascal matrix of first kind in base c and order n + 1. Then the following statements hold. (

a) P c [x] is an invertible matrix and its inverse is given by
can be factorized as follows

Generalized Apostol-Frobenius-Euler Polynomials
For m ∈ N, α, λ, u ∈ C and a, c ∈ R + , the generalized Apostol-type Frobenius-Euler polynomials in the variable x, parameters c, a, λ, order α and level m, are defined through the following generating function . For x = 0 we obtain, the generalized Apostol-Frobennius-Euler numbers of param- According to the Definition 3.1, with e = exp(1), we have (1.1) and (1.2) Example 3.2. For any λ ∈ C, m = 4, c = 2, a = 3, α = 1 and u = 2 the first the generalized Apostol-type Frobenius-Euler polynomials in the variable x, parameters c, a, λ, order α and level m are: (c) Differential relations. For m ∈ N and n, j ∈ N 0 with 0 ≤ j ≤ n, we have Proof. By substituting (2.1) into the right-hand side of (3.3) and using appropriate binomial coefficient identities (see, for instance [1,5,6]), we see that Therefore, (3.5) holds. Proof. By substituting (2.2) into the right-hand side of (3.3), it suffices to follow the proof given in Theorem 3.2, making the corresponding modifications. Proof. By substituting (2.3) into the right-hand side of (3.3), we see that Then, using appropriate combinational identities and summations (see, for instance [1,5,6]), we obtain n (x + y; c, a; λ; u) Therefore, (3.6) holds.   which implies the first equality of the theorem. The second and third equalities of can be derived in a similar way.