The generalized Bernoulli polynomials of order and the generalized Euler polynomials are defined by (see [3] ):

and

where denotes the set of complex numbers.

Recently, Luo and Srivastava [4] introduced the generalized Apostol-Bernoulli polynomials and the generalized Apostol-Euler polynomials as follows.

Definition 1.1. (Luo and Srivastava [4] ) The generalized Apostol-Bernoulli polynomials of order are defined by the generating function

Definition 1.2. (Luo [5] ) The generalized Apostol-Euler polynomials of order are defined by the generating function

Natalini and Bernardini [6] defined the new generalization of Bernoulli polynomials in the following definition.

Definition 1.3. The generalized Bernoulli polynomials, , are defined, in a suitable neighbourhood of by means of generating function

Recently, Tremblay et al. [7] investigated a new class of generalized Apostol-Bernoulli polynomial as follows.

Definition 1.4. The generalized Apostol-Bernoulli polynomials of order, , are defined, in a suitable neighbourhood of by means of generating function

Also, Sirvastava et al. [1] 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 and. Then the generalized Bernoulli polynomials of order are defined by the following generating function:

This sequel to the work by Sirvastava et al. [2] introduced and investigated a similar generalization of the family of Euler polynomials defined as follows.

Definition 1.6. Let and. Then the generalized Euler polynomials of order are defined by the following generating function

It is easy to see that setting and in (1.8) would lead to Apostol-Euler polynomials defined by (1.4). The case where has been studied by Luo et al. [8] .

In Section 2, we introduce the new extension of unified family of Apostol-type polynomials and numbers that are defined in [9] . Also, we determine relations between some results given in [1] [3] [7] [10] [11] and our results. Moreover, we introduce some new identities for polynomials defined in [9] . In Section 3, we give some basic properties of the new unification of Apostol-type polynomials and numbers. Finally in Section 4, we introduce some relationships between the new unification of Apostol-type polynomials and other known polynomials.

Definition 2.1. Let, and. Then the new unification of Apostol-type polynomials are defined, in a suitable neighbourhood of by means of generating function

where is a sequence of complex numbers.

Remark 2.1. If we set in (2.1), then we obtain the new unification of multiparameter Apostol-type numbers, as

The generating function in (2.1) gives many types of polynomials as special cases, for example, see Table 1.

Remark 2.2. From NO. 13 in Table 1 and ([9] , Table 1), we can obtain the polynomials and the numbers given in [12] -[16] .

Theorem 3.1. Let and. Then

Proof. For the first equation, from (2.1)

using Cauchy product rule, we can easily obtain (3.1).

For the second Equation (3.2), from (2.1)

Equating the coefficient of on both sides, yields (3.2).

Corollary 3.1. If in (3.1), we have

Theorem 3.2. The following identity holds true, when and in (2.1)

Proof. From (2.1)

Hence, we can easily obtain (3.5).

Remark 3.1. If we put, and in (3.5), then it gives [[12] , Equation (34)],

where is the unification of the Apostol-type polynomials.

Theorem 3.3. The unification of Apostol-type numbers satisfy

Proof. When in (2.1), we have

Using Cauchy product rule, we obtain (3.6).

Theorem 3.4. The following relationship holds true

where and and,.

Proof. Starting with (2.1), we get

Using Cauchy product rule on the right hand side of the last equation and equating the coefficients of on both sides, yields (3.7).

Using No. 13 in Table 1, we obtain Nörlund’s results, see [17] and Carlitz’s generalizations, see [18] by our approach in Theorem 3.5 and Theorem 3.6 as follows

Theorem 3.5. For, we have

Proof. For the first equation and starting with (2.1), we get

Equating the coefficients of on both sides, yields (3.8).

For the second equation and starting with (2.1), we get

then, we have

Equating coefficients of on both sides, yields (3.9).

Theorem 3.6. For and we have

Proof. For the first equation and starting with (2.1), we get

Equating the coefficients of on both sides, yields (3.10).

Also, It is not difficult to prove (3.11).

In this section, we give some relationships between the polynomials and Laguerre polynomials, Jacobi polynomials, Hermite polynomials, generalized Stirling numbers of second kind, Stirling numbers and Bleimann-Butzer-hahn basic.

Theorem 4.1. For, and, we have relationship

between the new unification of Apostol-type polynomials and generalized Stirling numbers of second kind, see [19] .

Proof. Using (3.4) and from definition of generalized Stirling numbers of second kind, we easily obtain (4.1).

Theorem 4.2. For, and, we have the relationship

between the new unification of Apostol-type polynomials and Stirling numbers of second kind.

Proof. Using (3.4) and from definition of Stirling numbers of second kind (see [20] ), we easily obtain (4.2).

Theorem 4.3. The relationship

holds between the new unification of multiparameter Apostol-type polynomials and generalized Laguerre polynomials (see [7] , No. (3), Table 1).

Proof. From (3.4) and substitute

then we get (4.3).

Theorem 4.4. For. The relationship

holds between the new unification of Apostol-type polynomials and Jacobi polynomials (see [21] , p. 49, Equation (35)).

Proof. From (3.4) and substitute

then we get (4.4).

Theorem 4.5. The relationship

holds between the new unification of Apostol-type polynomials and Hermite polynomials (see [7] , No. (1) Table 1).

Proof. From (3.4) and substitute

then we get (4.5).

Theorem 4.6. When, , and in (9) and for,

, , and, ,

, , we have the following relationship

between the new unified family of generalized Apostol-Euler, Bernoulli and Genocchi polynomials, and

(the generalized Lah numbers) (see [22] ).

Proof. From [9] , Equation (2.1),

Equating the coefficients of on both sides, yields (4.6).

Using No. 13 in Table 1 (see [9] ) and the definition of the unified Bernstein and Bleimann-Butzer-Hahn basis (see [23] ),

where, , , we obtain the following theorem.

Theorem 4.7. For we have relationship

between the unified Bernstein and Bleimann-Butzer-Hahn basis, the new unified family of generalized Apostol-Bernoulli, Euler and Genocchi polynomials (see [9] ) and generalized Stirling numbers of first kind (see [19] ).

Proof. From (2.1) and (4.7) and with some elementary calculation, we easily obtain (4.8).

Beih El-SayedEl-Desouky,Rabab SabryGomaa, (2015) New Extension of Unified Family Apostol-Type of Polynomials and Numbers. Applied Mathematics,06,1495-1505. doi: 10.4236/am.2015.69134