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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.

The generalized Bernoulli polynomials

and

where

Recently, Luo and Srivastava [

Definition 1.1. (Luo and Srivastava [

Definition 1.2. (Luo [

Natalini and Bernardini [

Definition 1.3. The generalized Bernoulli polynomials

Recently, Tremblay et al. [

Definition 1.4. The generalized Apostol-Bernoulli polynomials

Also, Sirvastava et al. [

Definition 1.5. Let

This sequel to the work by Sirvastava et al. [

Definition 1.6. Let

It is easy to see that setting

In Section 2, we introduce the new extension of unified family of Apostol-type polynomials and numbers that are defined in [

Definition 2.1. Let

where

Remark 2.1. If we set

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

Remark 2.2. From NO. 13 in

Theorem 3.1. Let

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)

1 | setting | |
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2 | setting | |

3 | setting | |

4 | setting | |

5 | setting | |

6 | setting | |

7 | setting | |

8 | setting | |

9 | setting | |

10 | setting | |

11 | setting | |

12 | setting | |

13 | setting |

Equating the coefficient of

Corollary 3.1. If

Theorem 3.2. The following identity holds true, when

Proof. From (2.1)

Hence, we can easily obtain (3.5).

Remark 3.1. If we put

where

Theorem 3.3. The unification of Apostol-type numbers satisfy

Proof. When

Using Cauchy product rule, we obtain (3.6).

Theorem 3.4. The following relationship holds true

where

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

Using No. 13 in

Theorem 3.5. For

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

Equating the coefficients of

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

then, we have

Equating coefficients of

Theorem 3.6. For

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

Equating the coefficients of

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

In this section, we give some relationships between the polynomials

Theorem 4.1. For

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

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

Theorem 4.2. For

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 [

Theorem 4.3. The relationship

holds between the new unification of multiparameter Apostol-type polynomials and generalized Laguerre polynomials (see [

Proof. From (3.4) and substitute

then we get (4.3).

Theorem 4.4. For

holds between the new unification of Apostol-type polynomials and Jacobi polynomials (see [

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 [

Proof. From (3.4) and substitute

then we get (4.5).

Theorem 4.6. When

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

Proof. From [

Equating the coefficients of

Using No. 13 in

where

Theorem 4.7. For

between the unified Bernstein and Bleimann-Butzer-Hahn basis, the new unified family of generalized Apostol-Bernoulli, Euler and Genocchi polynomials (see [

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