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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

References

  1. Jang L-C: A study on the distribution of twisted -Genocchi polynomials. Advanced Studies in Contemporary Mathematics 2009, 18(2):181–189.MathSciNetMATHGoogle Scholar
  2. Kim T: The modified -Euler numbers and polynomials. Advanced Studies in Contemporary Mathematics 2008, 16(2):161–170.MathSciNetMATHGoogle Scholar
  3. 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
  4. Kim T: -Volkenborn integration. Russian Journal of Mathematical Physics 2002, 9(3):288–299.MathSciNetMATHGoogle Scholar
  5. Kim T: On Euler-Barnes multiple zeta functions. Russian Journal of Mathematical Physics 2003, 10(3):261–267.MathSciNetMATHGoogle Scholar
  6. Kim T: Non-Archimedean -integrals associated with multiple Changhee -Bernoulli polynomials. Russian Journal of Mathematical Physics 2003, 10(1):91–98.MathSciNetMATHGoogle Scholar
  7. 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
  8. Kim T: -generalized Euler numbers and polynomials. Russian Journal of Mathematical Physics 2006, 13(3):293–298. 10.1134/S1061920806030058MathSciNetView ArticleMATHGoogle Scholar
  9. Kim T: -Bernoulli numbers and polynomials associated with Gaussian binomial coefficients. Russian Journal of Mathematical Physics 2008, 15(1):51–57.MathSciNetView ArticleMATHGoogle Scholar
  10. Kim T: A note on the generalized -Euler numbers. Proceedings of the Jangjeon Mathematical Society 2009, 12(1):45–50.MathSciNetMATHGoogle Scholar
  11. Kim T, Hwang K-W, Lee B: A note on the -Euler measures. Advances in Difference Equations 2009, 2009:-8.Google Scholar
  12. 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
  13. 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
  14. 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
  15. 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
  16. 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
  17. 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
  18. Carlitz L: -Bernoulli numbers and polynomials. Duke Mathematical Journal 1948, 15: 987–1000. 10.1215/S0012-7094-48-01588-9MathSciNetView ArticleMATHGoogle Scholar
  19. 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
  20. Cenkci M: The -adic generalized twisted -Euler--function and its applications. Advanced Studies in Contemporary Mathematics 2007, 15(1):37–47.MathSciNetMATHGoogle Scholar
  21. 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
  22. Govil NK, Gupta V: Convergence of -Meyer-König-Zeller-Durrmeyer operators. Advanced Studies in Contemporary Mathematics 2009, 19(1):97–108.MathSciNetMATHGoogle Scholar
  23. 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

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