- Research Article
- Open Access

- 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

- 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

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

## References

- Jang L-C: A study on the distribution of twisted -Genocchi polynomials.
*Advanced Studies in Contemporary Mathematics*2009, 18(2):181–189.MathSciNetMATHGoogle Scholar - Kim T: The modified -Euler numbers and polynomials.
*Advanced Studies in Contemporary Mathematics*2008, 16(2):161–170.MathSciNetMATHGoogle Scholar - Kim T: Symmetry of power sum polynomials and multivariate fermionic -adic invariant integral on .
*Russian Journal of Mathematical Physics*2009, 16(1):93–96. 10.1134/S1061920809010063MathSciNetView ArticleMATHGoogle Scholar - Kim T: -Volkenborn integration.
*Russian Journal of Mathematical Physics*2002, 9(3):288–299.MathSciNetMATHGoogle Scholar - Kim T: On Euler-Barnes multiple zeta functions.
*Russian Journal of Mathematical Physics*2003, 10(3):261–267.MathSciNetMATHGoogle Scholar - Kim T: Non-Archimedean -integrals associated with multiple Changhee -Bernoulli polynomials.
*Russian Journal of Mathematical Physics*2003, 10(1):91–98.MathSciNetMATHGoogle Scholar - Kim T: Power series and asymptotic series associated with the -analog of the two-variable -adic -function.
*Russian Journal of Mathematical Physics*2005, 12(2):186–196.MathSciNetMATHGoogle Scholar - Kim T: -generalized Euler numbers and polynomials.
*Russian Journal of Mathematical Physics*2006, 13(3):293–298. 10.1134/S1061920806030058MathSciNetView ArticleMATHGoogle Scholar - Kim T: -Bernoulli numbers and polynomials associated with Gaussian binomial coefficients.
*Russian Journal of Mathematical Physics*2008, 15(1):51–57.MathSciNetView ArticleMATHGoogle Scholar - Kim T: A note on the generalized -Euler numbers.
*Proceedings of the Jangjeon Mathematical Society*2009, 12(1):45–50.MathSciNetMATHGoogle Scholar - Kim T, Hwang K-W, Lee B: A note on the -Euler measures.
*Advances in Difference Equations*2009, 2009:-8.Google Scholar - Kim T: Note on the Euler -zeta functions.
*Journal of Number Theory*2009, 129(7):1798–1804. 10.1016/j.jnt.2008.10.007MathSciNetView ArticleMATHGoogle Scholar - Kim T: On a -analogue of the -adic log gamma functions and related integrals.
*Journal of Number Theory*1999, 76(2):320–329. 10.1006/jnth.1999.2373MathSciNetView ArticleMATHGoogle Scholar - Kim Y-H, Kim W, Ryoo CS: On the twisted -Euler zeta function associated with twisted -Euler numbers.
*Proceedings of the Jangjeon Mathematical Society*2009, 12(1):93–100.MathSciNetMATHGoogle Scholar - Ozden H, Cangul IN, Simsek Y: Remarks on -Bernoulli numbers associated with Daehee numbers.
*Advanced Studies in Contemporary Mathematics*2009, 18(1):41–48.MathSciNetMATHGoogle Scholar - Ozden H, Simsek Y, Rim S-H, Cangul IN: A note on -adic -Euler measure.
*Advanced Studies in Contemporary Mathematics*2007, 14(2):233–239.MathSciNetMATHGoogle Scholar - Rim S-H, Kim T: A note on -adic Euler measure on .
*Russian Journal of Mathematical Physics*2006, 13(3):358–361. 10.1134/S1061920806030113MathSciNetView ArticleMATHGoogle Scholar - Carlitz L: -Bernoulli numbers and polynomials.
*Duke Mathematical Journal*1948, 15: 987–1000. 10.1215/S0012-7094-48-01588-9MathSciNetView ArticleMATHGoogle Scholar - Cangul IN, Kurt V, Ozden H, Simsek Y: On the higher-order --Genocchi numbers.
*Advanced Studies in Contemporary Mathematics*2009, 19(1):39–57.MathSciNetMATHGoogle Scholar - Cenkci M: The -adic generalized twisted -Euler--function and its applications.
*Advanced Studies in Contemporary Mathematics*2007, 15(1):37–47.MathSciNetMATHGoogle Scholar - Can M, Cenkci M, Kurt V, Simsek Y: Twisted Dedekind type sums associated with Barnes' type multiple Frobenius-Euler -functions.
*Advanced Studies in Contemporary Mathematics*2009, 18(2):135–160.MathSciNetMATHGoogle Scholar - Govil NK, Gupta V: Convergence of -Meyer-König-Zeller-Durrmeyer operators.
*Advanced Studies in Contemporary Mathematics*2009, 19(1):97–108.MathSciNetMATHGoogle Scholar - Gupta V, Finta Z: On certain -Durrmeyer type operators.
*Applied Mathematics and Computation*2009, 209(2):415–420. 10.1016/j.amc.2008.12.071MathSciNetView ArticleMATHGoogle Scholar

## 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.