Interpolation Functions of -Extensions of Apostol's Type Euler Polynomials

Abstract

The main purpose of this paper is to present new -extensions of Apostol's type Euler polynomials using the fermionic -adic integral on . We define the --Euler polynomials and obtain the interpolation functions and the Hurwitz type zeta functions of these polynomials. We define -extensions of Apostol type's Euler polynomials of higher order using the multivariate fermionic -adic integral on . We have the interpolation functions of these --Euler polynomials. We also give -extensions of Apostol's type Euler polynomials of higher order and have the multiple Hurwitz type zeta functions of these --Euler polynomials.

1. Introduction, Definitions, and Notations

After Carlitz [1] gave -extensions of the classical Bernoulli numbers and polynomials, the -extensions of Bernoulli and Euler numbers and polynomials have been studied by several authors. Many authors have studied on various kinds of -analogues of the Euler numbers and polynomials (cf., [1–39]).T Kim [7–23] has published remarkable research results for -extensions of the Euler numbers and polynomials and their interpolation functions. In [13], T Kim presented a systematic study of some families of multiple -Euler numbers and polynomials. By using the -Volkenborn integration on , he constructed the -adic -Euler numbers and polynomials of higher order and gave the generating function of these numbers and the Euler --function. In [20], Kim studied some families of multiple -Genocchi and -Euler numbers using the multivariate -adic -Volkenborn integral on , and gave interesting identities related to these numbers. Recently, Kim [21] studied some families of -Euler numbers and polynomials of Nölund's type using multivariate fermionic -adic integral on .

Many authors have studied the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials, and their -extensions (cf., [1, 6, 25, 27, 28, 33–41]). Choi et al. [6] studied some -extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of order , and multiple Hurwitz zeta function. In [24], Kim et al. defined Apostol's type -Euler numbers and polynomials using the fermionic -adic -integral and obtained the generating functions of these numbers and polynomials, respectively. They also had the distribution relation for Apostol's type -Euler polynomials and obtained -zeta function associated with Apostol's type -Euler numbers and Hurwitz type -zeta function associated with Apostol's type -Euler polynomials for negative integers.

In this paper, we will present new -extensions of Apostol's type Euler polynomials using the fermionic -adic integral on , and then we give interpolation functions and the Hurwitz type zeta functions of these polynomials. We also give -extensions of Apostol's type Euler polynomials of higher order using the multivariate fermionic -adic integral on .

Let be a fixed odd prime number. Throughout this paper , , and will, respectively, denote the ring of -adic rational integers, the field of -adic rational numbers, the complex number field, and the completion of algebraic closure of . Let be the set of natural numbers and . Let be the normalized exponential valuation of with When one talks of -extension, is variously considered as an indeterminate, a complex number or a -adic number . If one normally assumes If then one assumes

Now we recall some -notations. The -basic natural numbers are defined by and the -factorial by . The -binomial coefficients are defined by

(1.1)

Note that , which is the binomial coefficient. The -shift factorial is given by

(1.2)

Note that . It is well known that the -binomial formulae are defined as

(1.3)

Since , it follows that

(1.4)

Hence it follows that

(1.5)

which converges to as

For a fixed odd positive integer with , let

(1.6)

where lies in . The distribution is defined by

(1.7)

Let be the set of uniformly differentiable functions on . For , the -adic invariant -integral is defined as

(1.8)

The fermionic -adic invariant -integral on is defined as

(1.9)

where . The fermionic -adic integral on is defined as

(1.10)

It follows that where . For , let . we have

(1.11)

For details, see [7–21].

The classical Euler numbers and the classical Euler polynomials are defined, respectively, as follows:

(1.12)

It is known that the classical Euler numbers and polynomials are interpolated by the Euler zeta function and Hurwitz type zeta function, respectively, as follows:

(1.13)

In Section 2, we define new -extensions of Apostol's type Euler polynomials using the fermionic -adic integral on which will be called the --Euler polynomials . Then we obtain the interpolation functions and the Hurwitz type zeta functions of these polynomials. In Section 3, we define -extensions of Apostol's type Euler polynomials of higher order using the multivariate fermionic -adic integral on . We have the interpolation functions of these higher-order --Euler polynomials. In Section 4, we also give -extensions of Apostol's type Euler polynomials of higher order and have the multiple Euler zeta functions of these --Euler polynomials.

2. -Extensions of Apostol's Type Euler Polynomials

First, we assume that with . In , the -Euler polynomials are defined by

(2.1)

and are called the -Euler numbers. Then it follows that

(2.2)

The generating functions of are defined as

(2.3)

By (2.3), the interpolation functions of the -Euler polynomials are obtained as follows:

(2.4)

Thus, we have the following theorem.

Theorem 2.1.

Assume with . Then one has

(2.5)

Differentiating at shows that

(2.6)

In , we assume that with . The -Euler polynomials are defined by

(2.7)

By (2.7), we have

(2.8)

For , the Hurwitz type zeta functions for the -Euler polynomials are given as

(2.9)

For , we have from (2.9) that

(2.10)

Now we give new -extensions of Apostol's type Euler polynomials. For , let be the cyclic group of order . Let be the p-adic locally constant space defined by

(2.11)

First, we assume that with . For , we define -Euler polynomials of Apostol's type using the fermionic -adic integral as follows:

(2.12)

and we will call them the --Euler polynomials. Then are defined as the --Euler numbers. From (2.12), we have

(2.13)

Let . From (2.12), we easily derive

(2.14)

On the other hand, we have

(2.15)

From (2.14) and (2.15), we obtain the following theorem.

Theorem 2.2.

Assume that with . For , let Then one has

(2.16)

In , we assume that with . Let with . We define the --Euler polynomials to be satisfied the following equation:

(2.17)

When we differentiate both sides of (2.17) at , we have

(2.18)

Hence we have the interpolation functions of the --Euler polynomials as follows:

(2.19)

For , we define the Hurwitz type zeta function of the --Euler polynomials as

(2.20)

where For , we have

(2.21)

3. -Extensions of Apostol's Type Euler Polynomials of Higher Order

In this section, we give the -extension of Apostol's type Euler polynomials of higher order using the multivariate fermionic -adic integral.

First, we assume that with . Let . We define the --Euler polynomials of order as follows:

(3.1)

Note that are called the --Euler number of order . Using the multivariate fermionic -adic integral, we obtain from (3.1) that

(3.2)

Let be the generating functions of defined by

(3.3)

By (2.12) and (3.3), we have

(3.4)

Thus we have the following theorem.

Theorem 3.1.

Assume that with . For and , let . Then one has

(3.5)

In , we assume that with and with for . We define the --Euler polynomial of order as follows:

(3.6)

From (3.6), we have

(3.7)

For , we define the multiple Hurwitz type zeta functions for --Euler polynomials as

(3.8)

where In the special case with , we have

(3.9)

4. -Extension of Apostol's Type Euler Polynomials of Higher Order

In this section, we give the -extension of --Euler polynomials of higher order using the multivariate fermionic -adic integral.

Assume that with . For , we define --Euler polynomials of order as follows:

(4.1)

Note that are called the --Euler numbers.

When , the --Euler polynomials are

(4.2)

where is the Gaussian binomial coefficient. From (4.2), we obtain the following theorem.

Theorem 4.1.

Assume that with . For and , let . Then one has

(4.3)

In , assume that with and with . Then we can define --Euler polynomials for as follows:

(4.4)

Differentiating both sides of (4.4) at , we have

(4.5)

From (4.5), we have

(4.6)

Then we have

(4.7)

For , we define the Hurwitz type zeta function of --Euler polynomials of order as

(4.8)

where

From (4.4) and (4.8), we easily see that

(4.9)