New Extension of Unified Family of Apostol-Type of Polynomials and Numbers

The purpose of this paper is to introduce and investigate a new unification of unified family of Apostol-type polynomials and numbers based on results given in [24] and [25]. Also, we derive some properties for these polynomials and obtain some relationships between the Jacobi polynomials, Laguerre polynomials, Hermite polynomials, Stirling numbers and some other types of generalized polynomials.

(1.4) Natalini and Bernardini [17] defined the new generalization of Bernoulli polynomials in the following form.
Also, Sirvastava et al. [24] introduced a new interesting class of Apostol-Bernoulli polynomials that are closely related to the new class that we present in this paper. They investigated the following form.
Definition 1.5. Let a, b, c ∈ R + (a = b)) and n ∈ N 0 . Then the generalized Bernoulli polynomials B (α) n (x; λ; a, b, c) of order α ∈ C are defined by the following generating function: In this sequel to the work by Sirvastava et al. [25] introduced and investigated a similar generalization of the family of Euler polynomials defined as follws.
Definition 1.6. Let a, b, c ∈ R + (a = b)) and n ∈ N 0 . Then the generalized Euler polynomials E (α) n (x; λ; a, b, c) of order α ∈ C are defined by the following generating function: It is easy to see that setting a = 1 and b = c = e in (1.8) would lead to Apostol-Euler polynomials defined by (1.4). The case where α = 1 has been studied by Luo et al. [12].
In Section 2, we introduce the new extension of unified family of Apostol-type polynomials and numbers that are defined in [7]. Also, we determine relation between some results given in [23,24,10,11,26] and our results and introduce some new identities for polynomials defined in [7] . In Section 3, we give some basic properties of the new unification of Apostoltype polynomials and numbers. Finally in Section 4, we introduce some relationships between the new unification of Apostol-type polynomials and other known polynomials. where k ∈ N 0 ; r ∈ C; α r = (α 0 , α 1 , ..., α r−1 ) is a sequence of complex numbers.  (unification of Apostol-type polynomials of order r, see [21]) (generalized Euler polynomials of order r, see [11])  13 in Table1 and [7,Table1], we can obtain the polynomials and the numbers given in [1,5,9,13,21]. )) and x∈R. Then

Some basic properties for the polynomial M
using Cauchy product rule, we can easily obtain (3.1). For the second equation Equating coefficient of t n n! on both sides, yields (3.2).
Theorem 3.4. The following relationship holds true
Proof. Starting with (2.1), we get Using Cauchy product rule on the right hand side of the last equation and equating coefficients of t n on both sides, yields (3.7).
Using No.13 in Table1, we obtain Nörlund , s results, see [19] and Carlitz , s generalizations, see [2] by our approach in Theorem 3.5 and Theorem 3.6 as follows  Proof. For the first equation and starting with (2.1), we get Equating coefficients of t ℓ on both sides, yields (3.8).
For the second equation and starting with (2.1), we get Equating coefficients of t ℓ on both sides, yields (3.9). Proof. For the first equation and starting with (2.1), we get Equating coefficients of t ℓ on both sides, yields (3.10). Also, It is not difficult to prove (3.11).   holds between the new unification of multiparameter Apostol-type polynomials and generalized Laguerre polynomials, see [26,No.(3) Table1].
Using No.13 in Table 1, see [7] and the definition of the unified Bernstein and Bleimann-Butzer-Hahn basis(see [18]), where k, m ∈ Z + , a, b ∈ R, t ∈ C, we obtain the following theorem