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

Beih El-Sayed El-Desouky^{}, Rabab Sabry Gomaa^{}

Department of Mathematics, Mansoura University, Mansoura, Egypt.

**DOI: **10.4236/am.2015.69134
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Department of Mathematics, Mansoura University, Mansoura, Egypt.

The purpose of this paper is to introduce and investigate new unification of unified family of Apostol-type polynomials and numbers based on results given in [1] [2]. 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.

Keywords

Euler, Bernoulli and Genocchi Polynomials, Stirling Numbers, Laguerre Polynomials, Hermite Polynomials

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El-Desouky, B. and Gomaa, R. (2015) New Extension of Unified Family Apostol-Type of Polynomials and Numbers. *Applied Mathematics*, **6**, 1495-1505. doi: 10.4236/am.2015.69134.

1. Introduction

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

(1.1)

and

(1.2)

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

(1.3)

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

(1.4)

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

(1.5)

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

(1.6)

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:

(1.7)

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

(1.8)

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.

2. Unification of Multiparameter Apostol-Type Polynomials and Numbers

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

(2.1)

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

(2.2)

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] .

3. Some Basic Properties for the Polynomial

Theorem 3.1. Let and. Then

(3.1)

(3.2)

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)

Table 1. Special cases.

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

Corollary 3.1. If in (3.1), we have

(3.3)

(3.4)

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

(3.5)

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

(3.6)

Proof. When in (2.1), we have

Using Cauchy product rule, we obtain (3.6).

Theorem 3.4. The following relationship holds true

(3.7)

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

(3.8)

(3.9)

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

(3.10)

(3.11)

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).

4. Some Relations between and Other Polynomials and Numbers

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

(4.1)

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

(4.2)

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

(4.3)

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

(4.4)

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

(4.5)

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

(4.6)

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] ),

(4.7)

where, , , we obtain the following theorem.

Theorem 4.7. For we have relationship

(4.8)

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).

Conflicts of Interest

The authors declare no conflicts of interest.

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