- Research Article
- Open Access

# Note on -Nasybullin's Lemma Associated with the Modified -Adic -Euler Measure

- Taekyun Kim
^{1}, - Young-Hee Kim
^{1}Email author, - Lee-Chae Jang
^{2}, - Seog-Hoon Rim
^{3}and - Byungje Lee
^{4}

**2010**:482717

https://doi.org/10.1155/2010/482717

© Taekyun Kim et al. 2010

**Received:**1 December 2009**Accepted:**14 March 2010**Published:**30 March 2010

## Abstract

We derive the modified -adic -measures related to -Nasybullin's type lemma.

## Keywords

- Euler Number
- Usual Convention

## 1. Introduction

Let be a fixed prime number. Throughout this paper, the symbols , , , and denote the ring of rational integers, the ring of -adic rational integers, the field of -adic rational numbers, and the completion of algebraic closure of , respectively. Let be the set of natural numbers and . The -adic absolute value in is normalized in such a way that (see [1–17]). For with , let be the least common multiple of and . We set

where lies in .

When one talks of -extension, is variously considered as an indeterminate, a complex number or a -adic number . In this paper, we assume that with (see [1–6, 18–23]). As the definition of -number, we use the following notations:

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

The -Euler numbers, , can be determined inductively by

with the usual convention of replacing by (see [11]). The modified -Euler numbers of are defined in [2] as follows:

with the usual convention of replacing by . For any positive integer ,

is known as a measure on (see [9]). In [2], the Witt's type formulas for are given by

The modified -Euler polynomials are also defined by

with the usual convention of replacing by (see [2]). Thus, we note that

Recently Govil and Gupta [22] have introduced a new type of q-integrated Meyer-König-Zeller-Durrmeyer (q-MKZD) operators, obtained moments for these operators, and estimated the convergence of these integrated q-MKZD operators. In this paper, we consider the q-extension which is in a direction different than that of Govil and Gupta [22].

Let be a field over . Then we call a function a -measure on if is finitely additive function defined on open-closed subsets in , whose values are in the field . Any open-closed subset in is a disjoint union of some finite intervals in , where is prime to , and therefore a -measure is determined by its values on all intervals in . Let denote the set of all rational numbers, whose denominator is a divisor of for some . In Section 2, we derive the modified -adic -measures related to -Nasybullin's type lemma.

## 2. The Modified -Adic -Measure

Let be a -valued function defined on with the following property.

There exist two constants such that

for any number . Suppose that is a root of the equation . Then we define

for any interval . From (2.2), we note that

Thus, we have

Therefore we obtain the following theorem.

Theorem 2.1.

For with and , let be a -valued function defined on with the following properties.

for any interval .

From (1.9), we note that

Let be the th -Euler polynomials and let be the th -Euler functions, that is, for ,

Note that is the Euler function. By (2.7), we see that

Thus, the -Euler function satisfies the properties of Theorem 2.1 with constants

Then is equal to , as reduces simply to . Therefore, we obtain the following theorem.

Theorem 2.2.

Then is a -measure on .

For with and , let be a primitive Dirichlet character modulo . Then the generalized -Euler numbers are defined as follows:

From (2.12) and (2.7), we can easily derive the following Witt's formula:

We can compute a -analogue of the -adic - -function by the following -adic -Mellin Mazur transform with respect to .

Let

Since the character is constant on the interval ,

where are the th generalized -Euler numbers attached to . For , we have

Assume that with . Let be the Teichmüller character mod . For , we set . Note that and are defined by for . For , we define

For (2.14), (2.16) and (2.17), we note that

Since for , we have . Let . Then we have

Therefore, we obtain the following theorem.

Theorem 2.3.

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.