#### Abstract

The main purpose of this paper is to present a systemic study of some
families of higher-order *q*-Euler numbers and polynomials with weight *α*. In particular, by
using the fermionic *p*-adic *q*-integral on , we give a new concept of *q*-Euler numbers
and polynomials with weight *α*.

#### 1. Introduction

Let be a fixed odd prime. 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 (see [1–14]). When one speaks of -extension, can be regarded as an indeterminate, complex number , or -adic number ; it is always clear from context. If , we assume . If , then we assume (see [1–14]).

In this paper, we use the notation of -number as follows: (see [1–14]). Note that for any with in the -adic case.

Let be the space of continuous functions on . For , the fermionic -adic -integral on is defined by (see [4–7]).

From (1.2), we note that where .

It is well known that the ordinary Euler polynomials are defined by with the usual convention of replacing by .

In the special case, and are called the th Euler numbers (see [1–14]).

By (1.5), we get the following recurrence relation as follows:

Recently, -Euler numbers are defined by with the usual convention about replacing by (see [1–12]).

Note that .

For , the weight -Euler numbers are also defined by with the usual convention about replacing by (see [4]).

The purpose of this paper is to present a systemic study of some families of higher-order -Euler numbers and polynomials with weight . In particular, by using the fermionic -adic -integral on , we give a new concept of -Euler numbers and polynomials with weight .

#### 2. Higher-Order -Euler Numbers and Polynomials with Weight

For , and , let us consider the expansion of higher-order -Euler polynomials with weight as follows: From (1.2) and (2.1), we note that: In the special case, , are called the higher-order -Euler numbers with weight .

By (2.1), we get

From (2.1), we can derive the following equation:

By (2.1), (2.2), (2.3), and (2.4), we see that

Therefore, we obtain the following theorem.

Theorem 2.1. *For and , one has
*

By simple calculation, we easily see that

#### 3. Polynomials

We now consider the polynomials (in ) by

By (3.1), we get

From (3.1) and (3.2), we can derive the following equation:

Therefore, by (3.2) and (3.3), we obtain the following theorem.

Theorem 3.1. *For and , one has
**
where and .*

Let with . Then we have

Thus, by (3.5), we obtain the following theorem.

Theorem 3.2. *For with , one has
**
Moreover,
*

By (3.1), we get where .

Thus, we note that

#### 4. Polynomials

Let us define polynomials as follows:

From (4.1), we have

By the calculation of the fermionic -adic -integral on , we see that

Thus, by (4.3), we obtain the following theorem.

Theorem 4.1. *For and , one has
*

It is easy to show that with the usual convention about replacing by .

From , we have

For in (4.7), we have

Therefore, by (4.8), we obtain the following theorem.

Theorem 4.2. *For and , one has
**
with the usual convention about replacing by .*

From the fermionic -adic -integral on , we easily get

By (4.1), we see that

Therefore, by (4.11), we obtain the following theorem.

Theorem 4.3. *For , , and , one has
**
In particular, for , one gets
*

Let with . Then one has

Therefore, by (4.14), we obtain the following theorem.

Theorem 4.4 (Multiplication formula). * For with , we have
*

#### 5. Polynomials and

In (2.1), we know that Thus, we get

Therefore, by (2.1) and (5.2), we obtain the following theorem.

Theorem 5.1. *For , and , one has
*

Note that

Therefore, by (5.4), we obtain the following theorem.

Theorem 5.2. *For , one has
*

Let with . Then we get

Therefore, by (5.6), we obtain the following theorem.

Theorem 5.3. *For with , one has
*

Let . Then we get

Therefore, by (5.8), we obtain the following theorem.

Theorem 5.4. *For , one has
*

Let in Theorem 5.4. Then we see that

From (4.6) and Theorem 5.1, we note that

It is easy to show that

By simple calculation, we get

From (5.13), we note that with the usual convention about replacing by .

Put in (5.11); we get

Thus, we have with the usual convention about replacing by .

#### Acknowledgment

This work was supported by the Dong-A University research fund.