Open Access

On the Identities of Symmetry for the -Euler Polynomials of Higher Order

Advances in Difference Equations20092009:273545

DOI: 10.1155/2009/273545

Received: 19 February 2009

Accepted: 18 June 2009

Published: 20 July 2009

Abstract

The main purpose of this paper is to investigate several further interesting properties of symmetry for the multivariate -adic fermionic integral on . From these symmetries, we can derive some recurrence identities for the -Euler polynomials of higher order, which are closely related to the Frobenius-Euler polynomials of higher order. By using our identities of symmetry for the -Euler polynomials of higher order, we can obtain many identities related to the Frobenius-Euler polynomials of higher order.

1. Introduction/Definition

Let be a fixed odd prime number. Throughout this paper, and will, respectively, denote the ring of -adic rational integer, the field of -adic rational numbers, the complex number field, and the completion of algebraic closure of . Let be the normalized exponential valuation of with . Let be the space of uniformly differentiable functions on . For , with , the fermionic -adic -integral on is defined as
(1.1)
(see [1]). Let us define the fermionic -adic invariant integral on as follows:
(1.2)
(see [18]). From (1.2), we have
(1.3)
(see [9, 10]), where . For with , let . Then, we define the -Euler numbers as follows:
(1.4)
where are called the -Euler numbers. We can show that
(1.5)
where are the Frobenius-Euler numbers. By comparing the coefficients on both sides of (1.4) and (1.5), we see that
(1.6)
Now, we also define the -Euler polynomials as follows:
(1.7)
In the viewpoint of (1.5), we can show that
(1.8)
where are the th Frobenius-Euler polynomials. From (1.7) and (1.8), we note that
(1.9)

(cf. [18, 1118]). For each positive integer , let . Then we have

(1.10)
The -Euler polynomials of order , denoted , are defined as
(1.11)

Then the values of at are called the -Euler numbers of order . When , the polynomials or numbers are called the -Euler polynomials or numbers. The purpose of this paper is to investigate some properties of symmetry for the multivariate -adic fermionic integral on . From the properties of symmetry for the multivariate -adic fermionic integral on , we derive some identities of symmetry for the -Euler polynomials of higher order. By using our identities of symmetry for the -Euler polynomials of higher order, we can obtain many identities related to the Frobenius-Euler polynomials of higher order.

2. On the Symmetry for the -Euler Polynomials of Higher Order

Let with (mod 2) and . Then we set
(2.1)
where
(2.2)
Thus, we note that this expression for is symmetry in and . From (2.1), we have
(2.3)
We can show that
(2.4)
By (1.4) and (1.11), we see that
(2.5)
Thus, we have
(2.6)
From (2.3), (2.4), and (2.5), we can derive
(2.7)
By the same method, we also see that
(2.8)

By comparing the coefficients on both sides of (2.7) and (2.8), we obtain the following.

Theorem 2.1.

For   with   , , and   , one has
(2.9)
Let and in (2.9). Then we have
(2.10)
From (2.10), we note that
(2.11)
If we take in (2.11), then we have
(2.12)
From (2.3), we note that
(2.13)

By the symmetric property of in , we also see that

(2.14)

By comparing the coefficients on both sides of (2.13) and (2.14), we obtain the following theorem.

Theorem 2.2.

For    with    and  , one has
(2.15)
Let and , we have
(2.16)
From (2.16), we can derive
(2.17)

Declarations

Acknowledgment

The present research has been conducted by the research grant of the Kwangwoon University in 2009.

Authors’ Affiliations

(1)
Division of General Education-Mathematics, Kwangwoon University
(2)
Department of Mathematics, Sogang University
(3)
Department of General Education, Kookmin University

References

  1. Kim T: Symmetry -adic invariant integral on for Bernoulli and Euler polynomials. Journal of Difference Equations and Applications 2008,14(12):1267–1277. 10.1080/10236190801943220MATHMathSciNetView ArticleGoogle Scholar
  2. Kim T: Note on the Euler numbers and polynomials. Advanced Studies in Contemporary Mathematics 2008,17(2):131–136.MATHMathSciNetGoogle Scholar
  3. Kim T: Note on -Genocchi numbers and polynomials. Advanced Studies in Contemporary Mathematics 2008,17(1):9–15.MATHMathSciNetGoogle Scholar
  4. Kim T: The modified -Euler numbers and polynomials. Advanced Studies in Contemporary Mathematics 2008,16(2):161–170.MATHMathSciNetGoogle Scholar
  5. 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.1999.2373MATHMathSciNetView ArticleGoogle Scholar
  6. Kim T: -Volkenborn integration. Russian Journal of Mathematical Physics 2002,9(3):288–299.MATHMathSciNetGoogle Scholar
  7. Kim T: -Bernoulli numbers and polynomials associated with Gaussian binomial coefficients. Russian Journal of Mathematical Physics 2008,15(1):51–57.MATHMathSciNetView ArticleGoogle Scholar
  8. Kim T, Choi JY, Sug JY: Extended -Euler numbers and polynomials associated with fermionic -adic -integral on . Russian Journal of Mathematical Physics 2007,14(2):160–163. 10.1134/S1061920807020045MATHMathSciNetView ArticleGoogle Scholar
  9. 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/S1061920809010063MATHMathSciNetView ArticleGoogle Scholar
  10. Kim T: On -adic interpolating function for -Euler numbers and its derivatives. Journal of Mathematical Analysis and Applications 2008,339(1):598–608. 10.1016/j.jmaa.2007.07.027MATHMathSciNetView ArticleGoogle Scholar
  11. Agarwal RP, Ryoo CS: Numerical computations of the roots of the generalized twisted -Bernoulli polynomials. Neural, Parallel & Scientific Computations 2007,15(2):193–206.MATHMathSciNetGoogle Scholar
  12. Cenkci M, Can M, Kurt V: -adic interpolation functions and Kummer-type congruences for -twisted and -generalized twisted Euler numbers. Advanced Studies in Contemporary Mathematics 2004,9(2):203–216.MATHMathSciNetGoogle Scholar
  13. Howard FT: Applications of a recurrence for the Bernoulli numbers. Journal of Number Theory 1995,52(1):157–172. 10.1006/jnth.1995.1062MATHMathSciNetView ArticleGoogle Scholar
  14. Kupershmidt BA: Reflection symmetries of -Bernoulli polynomials. Journal of Nonlinear Mathematical Physics 2005, 12: 412–422. 10.2991/jnmp.2005.12.s1.34MathSciNetView ArticleGoogle Scholar
  15. Ozden H, Simsek Y: Interpolation function of the -extension of twisted Euler numbers. Computers & Mathematics with Applications 2008,56(4):898–908. 10.1016/j.camwa.2008.01.020MathSciNetView ArticleGoogle Scholar
  16. Jang L-C: A study on the distribution of twisted -Genocchi polynomials. Advanced Studies in Contemporary Mathematics 2009,18(2):181–189.MATHMathSciNetGoogle Scholar
  17. Schork M: Ward's "calculus of sequences", -calculus and the limit . Advanced Studies in Contemporary Mathematics 2006,13(2):131–141.MATHMathSciNetGoogle Scholar
  18. Tuenter HJH: A symmetry of power sum polynomials and Bernoulli numbers. The American Mathematical Monthly 2001,108(3):258–261. 10.2307/2695389MATHMathSciNetView ArticleGoogle Scholar

Copyright

© Taekyun Kim et al. 2009

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.