- 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

## Keywords

## 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

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.

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.

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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