Abstract

The purpose of this paper is to investigate some properties of -Euler numbers and polynomials with weight 0. From those -Euler numbers with weight 0, we derive some identities on the -Euler numbers and polynomials with weight 0.

1. Introduction

Let be a fixed odd prime number. Throughout this paper , , and will denote the ring of -adic rational integers, the field of -adic rational numbers, and the completion of algebraic closure of . The -adic absolute value is defined by where for with and . In this paper, we assume that and with . As well-known definition, the Euler polynomials are defined by with the usual convention about replacing by (see [1–15]).

In this special case, , are called the th Euler numbers (see [1]). Recently, the -Euler numbers with weight are defined by with the usual convention about replacing by (see [3, 12]). The -number of is defined by (see [1–15]). Note that . Let us define the notation of -Euler numbers with weight 0 as . The purpose of this paper is to investigate some interesting identities on the -Euler numbers with weight 0.

2. On the Extended -Euler Numbers of Higher-Order with Weight 0

Let be the space of continuous functions on . For , the fermionic -adic -integral on is defined by Kim as follows: (see [1–12]). By (2.1), we get where and (see [4, 5]).

By (1.2), (2.1), and (2.2), we see that

In the special case, , we get where are the th Frobenius-Euler numbers. From (2.4), we note that the -Euler numbers with weight 0 are given by

Therefore, by (2.5), we obtain the following theorem.

Theorem 2.1. For , one has where are called the th Frobenius-Euler numbers.

Let us define the generating function of the -Euler numbers with weight 0 as follows:

Then, by (2.3) and (2.7), we get

Now we define the -Euler polynomials with weight 0 as follows:

Thus, (2.4) and (2.9), we get

From (2.10), we have where are called the th Frobenius-Euler polynomials (see [9]).

Therefore, by (2.11), we obtain the following theorem.

Theorem 2.2. For , one has where are called the th Frobenius-Euler polynomials.

From (2.2) and Theorem 2.2, we note that where with (mod 2).

Therefore, by (2.13), we obtain the following corollary.

Corollary 2.3. For , with (mod 2) and , one has

In particular, , we get , where and are called the th Euler numbers and polynomials which are defined by

By (2.2), we easily see that

Thus, by (2.16), we get

Therefore, by (2.16), we obtain the following theorem.

Theorem 2.4. For , one has

where are called the th Frobenius-Euler polynomials and are called the th Frobenius-Euler numbers. In particular, , we have where are called the th Euler numbers.

From (2.5) and Theorem 2.2, we note that where the usual convention about replacing by . By Theorems 2.2 and 2.4, we get

From (2.20) and (2.21), we have

For , by (2.20) and (2.22), we have

Therefore, by (2.23), we obtain the following theorem.

Theorem 2.5. For , one has

For , we have

Therefore, by (2.25), we obtain the following theorem.

Theorem 2.6. For , one has

From (2.20), we have

By Theorem 2.6 and (2.27), we get

Therefore, by (2.28), we obtain the following theorem.

Theorem 2.7. For , one has

Let be the space of continuous functions on . For , -adic analogue of Bernstein operator of order for is given by where (see [1, 6, 7]).

For , -adic Bernstein polynomial of degree is defined by (see [1, 6, 7]).

Let us take the fermionic -adic -integral on for one Bernstein polynomials in (2.31) as follows:

By simple calculation, we easily get

Therefore, by (2.32) and (2.33), we obtain the following theorem.

Theorem 2.8. For with , one has

In particular, , we get

By Theorems 2.1 and 2.2, we get where with .