International Journal of Mathematics and Mathematical Sciences

VolumeΒ 2012Β (2012), Article IDΒ 424189, 7 pages

http://dx.doi.org/10.1155/2012/424189

## Symmetry Fermionic -Adic -Integral on for Eulerian Polynomials

^{1}National Institute for Mathematical Sciences, Yuseong-daero 1689-gil, Yuseong-gu, Daejeon 305-811, Republic of Korea ^{2}Division of Cultural Education, Kyungnam University, Changwon 631-701, Republic of Korea

Received 18 June 2012; Accepted 14 August 2012

Academic Editor: CheonΒ Ryoo

Copyright Β© 2012 Daeyeoul Kim and Min-Soo Kim. 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.

#### Abstract

Kim et al. (2012) introduced an interesting *p*-adic analogue of the
Eulerian polynomials. They studied some identities on the Eulerian polynomials in
connection with the Genocchi, Euler, and tangent numbers. In this paper, by
applying the symmetry of the fermionic *p*-adic *q*-integral on , defined by
Kim (2008), we show a
symmetric relation between the *q*-extension of the alternating sum of integer
powers and the Eulerian polynomials.

#### 1. Introduction

The Eulerian polynomials , which can be defined by the generating function have numerous important applications in number theory, combinatorics, and numerical analysis, among other areas. From (1.1), we note that where is the Kronecker symbol (see [1]). Thus far, few recurrences for the Eulerian polynomials other than (1.2) have been reported in the literature. Other recurrences are of importance as they might reveal new aspects and properties of the Eulerian polynomials, and they can help simplify the proofs of known properties. For more important properties, see, for instance, [1] or [2].

Let be a fixed odd prime number. Let , and be the ring of -adic integers, the field of -adic numbers, and the completion of the algebraic closure of , respectively. Let be the -adic valuation on , where . The extended valuation on is denoted by the same symbol . Let be an indeterminate, where . Then, the -number is defined by

For a uniformly (or strictly) differentiable function (see [1, 3β6]), the fermionic -adic -integral on is defined by Then, it is easy to see that where .

By using the same method as that described in [1], and applying (1.5) to , where for , we consider the generalized Eulerian polynomials on by using the fermionic -adic -integral on as follows: By expanding the Taylor series on the left-hand side of (1.7) and comparing the coefficients of the terms , we get

We note that, by substituting into (1.8), is the Witt's formula for the Eulerian polynomials in [1, Theorem 1]. Recently, Kim et al. [1] investigated new properties of the Eulerian polynomials at associated with the Genocchi, Euler, and tangent numbers.

Let denote the -extension of the alternating sum of integer powers, namely, where . If , is the alternating sum of integer powers (see [4]). In particular, we have

Let be any positive odd integers. Our main result of symmetry between the -extension of the alternating sum of integer powers and the Eulerian polynomials is given in the following theorem, which is symmetric in and .

Theorem 1.1. *Let be any positive odd integers and . Then, one has
*

Observe that Theorem 1.1 can be obtained by the same method as that described in [4]. If , Theorem 1.1 reduces to the form stated in the remark in [4, page 1275].

Using (1.11), if we take in Theorem 1.1, we obtain the following corollary.

Corollary 1.2. *Let be any positive odd integer and . Then, one has
*

#### 2. Proof of Theorem 1.1

For the proof of Theorem 1.1, we will need the following two identities (see (2.4) and (2.5)) related to the Eulerian polynomials and the -extension of the alternating sum of integer powers.

Let be any positive odd integers. From (1.7), we obtain This has an interesting -adic analytic interpretation, which we shall discuss below (see Remark 2.1). It is easy to see that the right-hand side of (2.1) can be written as In (1.10), let . The left-hand, right-hand side, by definition, becomes A comparison of (2.1) and (2.3) yields the identity By slightly modifying the derivation of (2.4), we can obtain the following identity:

*Remark 2.1. *The derivations of identities are based on the fermionic -adic -integral expression of the generating function for the Eulerian polynomials in (1.7) and the quotient of integrals in (2.4), (2.5) that can be expressed as the exponential generating function for the -extension of the alternating sum of integer powers.

Observe that similar identities related to the Eulerian polynomials and the -extension of the alternating sum of integer powers in (2.4) and (2.5) can be found, for instance, in [3, ], [4, ], and [6, Theorem 4].

* Proof of Theorem 1.1. *Let be any positive odd integers. Using the iterated fermionic -adic -integral on and (1.7), we have
Now, we put
From (1.7) and (2.5), we see that
On the other hand, from (1.7) and (2.4), we have
By comparing the coefficients on both sides of (2.8) and (2.9), we obtain the result in Theorem 1.1.

#### 3. Concluding Remarks

Note that many other interesting symmetric properties for the Euler, Genocchi, and tangent numbers are derivable as corollaries of the results presented herein. For instance, considering [1, ], where denotes the th Euler number defined by , and the Euler polynomials are defined by the generating function and on putting in Theorem 1.1 and Corollary 1.2, we obtain These formulae are valid for any positive odd integers . The Genocchi numbers may be defined by the generating function which have several combinatorial interpretations in terms of certain surjective maps on finite sets. The well-known identity shows the relation between the Genocchi and the Bernoulli numbers. It follows from (3.6) and the Staudt-Clausen theorem that the Genocchi numbers are integers. It is easy to see that and from (3.2), (3.5) we deduce that It is well known that the tangent coefficients (or numbers) , defined by are closely related to the Bernoulli numbers, that is, (see [1]) Ramanujan ([7, page 5]) observed that and, therefore, the tangent coefficients, are integers for . From (3.3), (3.6), (3.7), and (3.10), the obtained symmetric formulae involve the Bernoulli, Genocchi, and tangent numbers (see [1]).

#### Acknowledgment

This work was supported by the Kyungnam University Foundation grant, 2012.

#### References

- D. S. Kim, T. Kim, W. J. Kim, and D. V. Dolgy, βA note on Eulerian polynomials,β
*Abstract and Applied Analysis*, vol. 2012, Article ID 269640, 10 pages, 2012. View at Publisher Β· View at Google Scholar - P. Luschny, βEulerian polynomials,β http://www.luschny.de/math/euler/EulerianPolynomials.html, 2011.
- D. S. Kim and K. H. Park, βIdentities of symmetry for Euler polynomials arising from quotients of fermionic integrals invariant under ${S}_{3}$,β
*Journal of Inequalities and Applications*, vol. 2010, Article ID 851521, 16 pages, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - T. Kim, βSymmetry $p$-adic invariant integral on ${\mathbb{Z}}_{p}$ for Bernoulli and Euler polynomials,β
*Journal of Difference Equations and Applications*, vol. 14, no. 12, pp. 1267β1277, 2008. View at Publisher Β· View at Google Scholar - C. S. Ryoo, βA note on $q$-Bernoulli numbers and polynomials,β
*Applied Mathematics Letters*, vol. 20, no. 5, pp. 524β531, 2007. View at Publisher Β· View at Google Scholar - Y. Simsek, βComplete sum of products of $(h,q)$-extension of Euler polynomials and numbers,β
*Journal of Difference Equations and Applications*, vol. 16, no. 11, pp. 1331β1348, 2010. View at Publisher Β· View at Google Scholar - S. Ramanujan,
*Collected Papers of Srinivasa Ramanujan*, AMS Chelsea Publishing, Providence, RI, USA, 2000.