Some New Classes of Generalized Apostol-Euler and Apostol-Genocchi Polynomials

The main object of this paper is to introduce and investigate two new classes of generalized Apostol-Euler and Apostol-Genocchi polynomials. In particular, we obtain a new addition formula for the new class of the generalized Apostol-Euler polynomials. We also give an extension and some analogues of the Srivastava-Pintér addition theorem obtained in the works by Srivastava and Pintér 2004 and R. Tremblay, S. Gaboury, B.-J. Fugère, and Tremblay et al. 2011 . for both classes.


Introduction
The generalized Bernoulli polynomials B α n x of order α ∈ C, the generalized Euler polynomials E α n x of order α ∈ C, and the generalized Genocchi polynomials G α n x of order α ∈ C, each of degree n as well as in α, are defined, respectively, by the following generating functions see, 1, volume 3, page 253 et seq., 2, Section 2.8 , and 3 : |t| < π; 1 α : 1 .

International Journal of Mathematics and Mathematical Sciences
The literature contains a large number of interesting properties and relationships involving these polynomials 1, 4-7 .These appear in many applications in combinatorics, number theory, and numerical analysis.
Many interesting extensions to these polynomials have been given.In particular, Luo and Srivastava 8, 9 introduced the generalized Apostol-Bernoulli polynomials B α n x; λ of order α ∈ C; Luo 10 invented the generalized Apostol-Euler polynomials E α n x; λ of order α ∈ C and the generalized Apostol-Genocchi polynomials G α n x; λ of order α ∈ C in 3 .These polynomials are defined, respectively, as follows.
Definition 1.1.The generalized Apostol-Bernoulli polynomials B α n x; λ of order α ∈ C are defined by means of the following generating function: Definition 1.2.The generalized Apostol-Euler polynomials E α n x; λ of order α are defined by means of the following generating function: Definition 1.3.The generalized Apostol-Genocchi polynomials G α n x; λ of order α are defined by means of the following generating function: Recently, the authors 21 studied a new family of generalized Apostol-Bernoulli polynomials of order α in the following form.
It easy to see that if we set m 1, b c e in 1.8 , we arrive at the following: This is the generating function for the generalized Apostol-Bernoulli polynomials of order α.

1.13
It is easy to see that setting a 1 and b c e in 1.12 and 1.13 yields the classical results for the Apostol-Euler and Apostol-Genocchi polynomials.
Lately, Kurt 23  x , m ∈ N, are defined, in a suitable neighborhood of t 0 by means of the generating function: It is easy to see that if we set m 1 in 1.14 , we arrive at the following: which is the generating function for the generalized Euler polynomials of order α.Thus, we have In this paper, we propose a further generalization of Apostol-Euler polynomials and the Apostol-Genocchi polynomials and we give some properties involving them.For the new class of Apostol-Euler polynomials, we establish a new addition theorem with the help of a result given by Srivastava et al. 24 .We also give an extension of the Srivastava-Pintér theorem 25, 26 .Finally, we exhibit some relationships between the generalized Apostol-Euler polynomials and other polynomials or special functions with the help of the new addition formula.

New Classes of Generalized Apostol-Euler and Apostol-Genocchi Polynomials
The following definitions provide a natural generalization of the Apostol-Euler polynomials It is easy to see that if we set m 1, b c e in 2.1 , we arrive at the following: This is the generating function for the generalized Apostol-Euler polynomials of order α.Thus, we have x, e, e; λ E α n x; λ .

2.3
Definition 2.2.For arbitrary real or complex parameter α and for b, c ∈ R , the generalized Apostol-Genocchi polynomials Obviously, if we set m 1, b c e in 2.4 , we obtain This is the generating function for the generalized Apostol-Genocchi polynomials of order α.Thus, we have x, e, e; λ G α n x; λ .

2.6
The generalized Apostol-Euler polynomials x, b, c; λ defined by 2.1 possess the following interesting properties.These are stated as Theorems 2.3, 2.4, and 2.5 below.
International Journal of Mathematics and Mathematical Sciences or, equivalently, by Proof.Considering the generating function 2.1 , the relations 2.7 and 2.8 follow easily.x, b, c; λ satisfy the following relations: Proof.Considering the generating function 2.1 and comparing the coefficients of t n /n! in the both sides of the above equation, we arrive at 2.9 .Proof of 2.10 is similar.
Theorem 2.5.The generalized Apostol-Euler polynomials x, b, c; λ satisfy the following recurrence relation: x, b, c; λ and using the generating functions 2.1 and 1.12 , 2.11 follows easily.
Remark 2.6.Setting m 1 and b c e in 2.11 and with the help of 2.3 , we find Using the well-known result see 8 2.12 becomes the familiar relation for the generalized Apostol-Euler polynomials see 8 : x; λ .

2.14
International Journal of Mathematics and Mathematical Sciences 7 Now, let us shift our focus on some interesting properties for the generalized Apostol-Genocchi polynomials G m−1,α n x, b, c; λ defined by 2.4 .These are stated as Theorems 2.7, 2.8, and 2.9 below.

2.15
Proof.Considering the generating function 2.4 , the relations 2.15 follow easily.x, b, c; λ satisfy the following relations: x, b, c; λ y log c n−k .

2.17
Proof .Considering the generating function 2.4 and comparing the coefficients of t n /n! in the both sides of the above equation, we arrive at 2.17 .Proof of 2.18 is similar.
Remark 2.10.Putting m 1 and b c e in 2.19 and with the help of 2.6 , we find

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Using the well-known result see 11 2.20 becomes the familiar relation for the generalized Apostol-Genocchi polynomials see 11 :

An Addition Theorem for the New Class of Generalized Apostol-Euler Polynomials
In this section, we establish a new addition theorem for the generalized Apostol-Euler polynomials.This new formula is based on a result due to Srivastava et al. 24 .
The next theorem has been invented by Srivastava et al. 24 .However, the theorem is given without proof see 24, pages 438-440 .Theorem 3.1.Let B z and z −1 C z be arbitrary functions which are analytic in the neighborhood of the origin, and assume (for sake of simplicity) that B 0 C 0 1.

3.1
Define the sequence of functions {f where α and x are arbitrary complex numbers independent of z.Then, for arbitrary parameters σ and y, Remark 3.2.The choice of 1 in the conditions of 3.3 is merely a convenient one.In fact, any nonzero constant values may be assumed for B 0 and C 0 .Now, applying the last theorem with special choices of functions and parameters furnishes the next very interesting addition formula.This formula is contained in the following corollary.Moreover, if we set σ 0 in 3.4 , we obtain

3.5
This result 3.5 will be very useful in the next section.

Some Analogues of the Srivastava-Pint ér Addition Theorem
In this section, we give a generalization of the Srivastava-Pintér addition theorem and an analogue.We end this section by giving two interesting relationships involving the new addition formula 3.5 .
Theorem 4.1.The following relationship, holds between the new class of generalized Apostol-Euler polynomials, the classical Apostol-Euler polynomials defined by 1.4 , and the generalized Apostol-Euler polynomials defined by 1.12 .
Gegenbauer polynomials 27, page 283, 36 Stirling numbers of the second kind Proof.First of all, if we substitute the entry 9 for x n from Table 1 into the right-hand side of 2.10 , we get x y, b, c; λ International Journal of Mathematics and Mathematical Sciences 11 which, upon inverting the order of summation and using the following elementary combinatorial identity:

4.4
The innermost sum in 4.4 can be calculated with the help of 2.10 with, of course, We thus find from 4.4 that  Making use of Table 1 that contains a list of series representation for x n in terms of special polynomials or numbers, we can find some analogues of the Srivastava-Pintér addition theorem.Let us give an example of such formula.H n x t n .4.9 Proof.We derived the Proof from the addition theorem 2.10 and entry 1.
We end this paper by giving two special cases of the addition theorem 3.4 involving the new class of generalized Apostol-Euler polynomials.These are contained in the two next theorems.Proof.We derived the Proof from the addition theorem 3.5 and entry 5. Proof.We derived the Proof from the addition theorem 3.5 and entry 12.
It could be interesting to apply the addition formula 3.3 to other family of polynomials in conjunction with series representation involving some special functions for x n in order to derive some analogues of the Srivastava-Pintér addition theorem.

Definition 1 . 4 .
For arbitrary real or complex parameter α and for b, c ∈ R , the generalized Apostol-Bernoulli polynomials B m−1,α n x, b, c; λ , m ∈ N, λ ∈ C, are defined, in a suitable neighborhood of t 0, with |t log b| < 2π if λ 1 or with |t log b| < | log λ| if λ / 1, by means of the generating function:

Definition 2 . 1 .
For arbitrary real or complex parameter α and for b, c ∈ R , the generalized Apostol-Euler polynomials E m−1,α n x, b, c; λ , m ∈ N, λ ∈ C, are defined, in a suitable neighborhood of t 0, with |t log b| < | log −λ | by means of the generating function:

Corollary 3 . 3 .t
Let b, c ∈ R and m ∈ N.Then, for arbitrary complex parameters α, σ, x and y, the generalized Apostol-Euler polynomials E m−1,α n x, b, c; λ satisfy the addition formula: log b l /l! and C z t log c in Theorem 3.1, the result follows.

4 . 7 International
Journal of Mathematics and Mathematical Sciencesholds between the new class of generalized Apostol-Genocchi polynomials, the classical Apostol-Euler polynomials defined by 1.4 , and the generalized Apostol-Genocchi polynomials defined by 1.13 .

Theorem 4 . 3 .
The following relationship, between the new class of generalized Apostol-Euler polynomials and the Hermite polynomials defined bye 2xt−t 2 ∞ n 0

Theorem 4 . 4 . 5 ]
The following relationship, new class of generalized Apostol-Euler polynomials and the Stirling numbers of the second kind that could be computed by the formula[30, page 58, 1.

Theorem 4 . 5 .
The following relationship, n − k − l !S l j, j; λ B j n−k−l y; λ j ∈ N 0 , 4.12 holds between the new class of generalized Apostol-Euler polynomials and the classical Apostol-Bernoulli polynomials and the generalized Stirling numbers.
Srivastava et al. 22introduced two new families of generalized Euler and Genocchi polynomials.They investigated the following forms.
presented a new interesting class of generalized Euler polynomials.Explicitly, he introduced the next definition.

Table 1 :
x n expressed in terms of sums of special polynomials or numbers.
6 which, with the relation 2.11 , leads us to the relationship 4.7 asserted by Theorem 4.1.