On the Higher-Order q-Euler Numbers and Polynomials with Weight α

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) and (2.2), we have

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

By (4.3) and (4.6), we get

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