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  • Research Article
  • Open Access

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

  • 1,
  • 1Email author,
  • 2,
  • 3 and
  • 4
Journal of Inequalities and Applications20102010:482717

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

  • Received: 1 December 2009
  • Accepted: 14 March 2010
  • Published:

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 [117]). For with , let be the least common multiple of and . We set

(1.1)

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 [16, 1823]). As the definition of -number, we use the following notations:

(1.2)

(see [123]).

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

(1.3)

(see [2, 3]).

The -Euler numbers, , can be determined inductively by

(1.4)

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

(1.5)

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

(1.6)

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

(1.7)

The modified -Euler polynomials are also defined by

(1.8)

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

(1.9)

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

(2.1)

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

(2.2)

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

(2.3)

Thus, we have

(2.4)

Therefore we obtain the following theorem.

Theorem 2.1.

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

There exist two constants such that
(2.5)
for any . Suppose that is a root of the equation . Then there exists a -measure on such that
(2.6)

for any interval .

From (1.9), we note that

(2.7)

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

(2.8)

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

(2.9)

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

(2.10)

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

Theorem 2.2.

For , let the function be defined on as follows:
(2.11)

Then is a -measure on .

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

(2.12)

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

(2.13)

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

Let

(2.14)

Since the character is constant on the interval ,

(2.15)

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

(2.16)

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

(2.17)

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

(2.18)

Since for , we have . Let . Then we have

(2.19)

Therefore, we obtain the following theorem.

Theorem 2.3.

For , we have
(2.20)

Authors’ Affiliations

(1)
Division of General Education-Mathematics, Kwangwoon University, Seoul, 139-701, South Korea
(2)
Department of Mathematics and Computer Science, KonKuk University, Chungju, 380-701, South Korea
(3)
Department of Mathematics Education, Kyungpook National University, Taegu, 702-701, South Korea
(4)
Department of Wireless Communications Engineering, Kwangwoon University, Seoul, 139-701, South Korea

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Copyright

© Taekyun Kim et al. 2010

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.

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