Abstract and Applied Analysis

Volume 2012 (2012), Article ID 269640, 10 pages

doi:10.1155/2012/269640

Research Article

A Note on Eulerian Polynomials

D. S. Kim,¹ T. Kim,² W. J. Kim,³ and D. V. Dolgy⁴

¹Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea

²Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea

³Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea

⁴Hanrimwon, Kwangwoon University, Seoul 139-701, Republic of Korea

Received 29 May 2012; Accepted 25 June 2012

Academic Editor: Josef Diblík

Copyright © 2012 D. S. Kim et al. 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.

We study Genocchi, Euler, and tangent numbers. From those numbers we derive some identities on Eulerian polynomials in connection with Genocchi and tangent numbers.

1. Introduction

As is well known, the Eulerian polynomials, , are defined by generating function as follows:

with the usual convention about replacing by (see [1–18]). From (1.1), we note that

where is the Kronecker symbol (see [3]).

Thus, by (1.2), we get

By (1.1), (1.2), and (1.3), we see that

where and (see [1]).

The Genocchi polynomials are defined by

(see [6–18]). In the special case, , are called the th Genocchi numbers (see [14, 17, 18]).

It is well known that the Euler polynomials are also defined by

(see [1–5, 19–24]). Here , then is called the th Euler number. From (1.6), we have

(see [3–5, 19–23]).

As is well known, the Bernoulli numbers are defined by

(see [5, 18, 19]), with the usual convention about replacing by .

From (1.8), we note that the Bernoulli polynomials are also defined as

(see [5, 18, 19]).

The tangent numbers are defined as the coefficients of the Taylor expansion of :

(see [1–3, 5]).

In this paper, we give some identities on the Eulerian polynomials at associated with Genocchi, Euler, and tangent numbers.

2. Witt's Formula for Eulerian Polynomials

In this section, we assume that , , and will, respectively, denote the ring of -adic integers, the field of -adic numbers, and the completion of algebraic closure of . The -adic norm is normalized so that .

Let be an indeterminate with . Then the number is defined by

(see [6–18]).

Let be the space of continuous functions on . For , the fermionic -adic -integral on is defined by

(see [7, 10–13]). From (2.2), we can derive the following:

where .

Let us take . Then, by (2.3), we get

Thus, from (2.4), we have

By Taylor expansion on the left-hand side of (2.5), we get

Comparing coefficients on the both sides of (2.6), we have

Therefore, by (2.7), we obtain the following theorem.

Theorem 2.1. For , one has

where is an Eulerian polynomials.

It seems interesting to study Theorem 2.1 at . By (2.3), we get

where . From (2.9), we can derive the following equation:

where (see [5–13]).

From (2.9), we can derive the following:

By (2.11), we get

From (1.10) and (2.12), we have

By comparing coefficients on the both sides of (2.13), we get

where is the th tangent number.

Therefore, by (2.14), we obtain the following theorem.

Theorem 2.2. For , one has

where is the th tangent numbers.

From Theorem 2.1, one has

Therefore, by Theorem 2.2 and (2.16), we obtain the following corollary.

Corollary 2.3. For , one has

From (1.6) and (2.9), we have

(see [5]). Thus, by (2.16) and (2.18), we get

Therefore, by Corollary 2.3 and (2.19), we obtain the following corollary.

Corollary 2.4. For , one has

By (1.5) and (2.9), we get

By (2.21), we get

Thus, from (2.19), Theorem 2.2 and Corollary 2.3, we have

Therefore, by (2.23), we obtain the following theorem.

Theorem 2.5. For , one has

From (1.5), we note that

(see [13, 14]). Thus, by (2.25), we get

(see [13, 14]), with the usual convention about replacing by .

From (1.5) and (2.9), one has

Thus, by (2.27), we get

From (2.28), we have

Therefore, by (2.19), Corollary 2.3 and (2.29), we obtain the following theorem.

Theorem 2.6. For , we have

In particular,

3. Further Remark

In complex plane, we note that

By (1.10) and (3.1), we also get

From (1.5), we have

Thus, by (1.10) and (3.3), we get

From (3.4), we have

By (1.1), we see that

Thus, we note that

From (3.7), we have

It is easy to show that

For simple calculation, we can derive the following equation:

By (3.10), we get

Thus, from (3.11),we have

By (1.10) and (3.12), we get

From Corollary 2.3 and (3.13), we can derive the following identity:

Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology 2012R1A1A2003786. Also, the authors would like to thank the referees for their valuable comments and suggestions.

References

L. Euler, Institutiones calculi differentialis cum eius usu in analysi finitorum ac Doctrina serierum, Methodus summanandi superior ulterius pro-mota, chapter VII, Academiae Imperialis ScientiarumPetropolitanae, St Petersbourg, Russia, 1755.

D. Foata, “Eulerian polynomials: from Euler's time to the present,” in The legacy of Alladi Ramakrishnan in the Mathematical Sciences, pp. 253–273, Springer, New York, NY, USA, 2010.

D. S. Kim, T. Kim, Y. H. Kim, and D. V. Dolgy, “A note on Eulerian polynomi-als associated with Bernoulli and Euler numbers and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 22, no. 3, pp. 342–353, 2012.

D. S. Kim, T. Kim, S.-H. Lee, D. V. Dolgy, and S.-H. Rim, “Some new identities on the Bernoulli and Euler numbers,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 856132, 11 pages, 2011.

D. S. Kim, D. V. Dolgy, T. Kim, and S.-H. Rim, “Some formulae for the product of two Bernoulli and Euler polynomials,” Abstract and Applied Analysis, vol. 2012, Article ID 784307, 15 pages, 2012.

T. Kim, “Euler numbers and polynomials associated with zeta functions,” Abstract and Applied Analysis, vol. 2008, Article ID 581582, 11 pages, 2008.

T. Kim, “An identity of the symmetry for the Frobenius-Euler polynomials associated with the Fermionic -adic invariant -integrals on ,” The Rocky Mountain Journal of Mathematics, vol. 41, no. 1, pp. 239–247, 2011.

T. Kim, “Some identities on the -Euler polynomials of higher order and -Stirling numbers by the fermionic -adic integral on ,” Russian Journal of Mathematical Physics, vol. 16, no. 4, pp. 484–491, 2009.

T. Kim, “Symmetry of power sum polynomials and multivariate fermionic -adic invariant integral on ,” Russian Journal of Mathematical Physics, vol. 16, no. 1, pp. 93–96, 2009.

H. Ozden, I. N. Cangul, and Y. Simsek, “Multivariate interpolation functions of higher-order -Euler numbers and their applications,” Abstract and Applied Analysis, vol. 2008, Article ID 390857, 16 pages, 2008.

H. Ozden, I. N. Cangul, and Y. Simsek, “Remarks on -Bernoulli numbers associated with Daehee numbers,” Advanced Studies in Contemporary Mathematics, vol. 18, no. 1, pp. 41–48, 2009.

S.-H. Rim and S.-J. Lee, “Some identities on the twisted -Genocchi numbers and polynomials associated with -Bernstein polynomials,” International Journal of Mathematics and Mathematical Sciences, vol. 2011, Article ID 482840, 8 pages, 2011.

S.-H. Rim, J.-H. Jin, E.-J. Moon, and S.-J. Lee, “Some identities on the -Genocchi polynomials of higher-order and -Stirling numbers by the fermionic -adic integral on ,” International Journal of Mathematics and Mathematical Sciences, vol. 2010, Article ID 860280, 14 pages, 2010.

S.-H. Rim, K. H. Park, and E. J. Moon, “On Genocchi numbers and polynomials,” Abstract and Applied Analysis, vol. 2008, Article ID 898471, 7 pages, 2008.

C. S. Ryoo, “Some relations between twisted -Euler numbers and Bernstein polynomials,” Advanced Studies in Contemporary Mathematics, vol. 21, no. 2, pp. 217–223, 2011.

J. P. O. Santos, “On a new combinatorial interpretation for a theorem of Euler,” Advanced Studies in Contemporary Mathematics, vol. 3, no. 2, pp. 31–38, 2001.

Y. Simsek, “Complete sum of products of -extension of Euler polynomials and numbers,” Journal of Difference Equations and Applications, vol. 16, no. 11, pp. 1331–1348, 2010.

Y. Simsek, “Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions,” Advanced Studies in Contemporary Mathematics, vol. 16, no. 2, pp. 251–278, 2008.

S. Araci, D. Erdal, and J. J. Seo, “A study on the fermionic -adic -integral representation on associated with weighted -Bernstein and -Genocchi polynomials,” Abstract and Applied Analysis, vol. 2011, Article ID 649248, 10 pages, 2011.

A. A. Aygunes and Y. Simsek, “Unification of multiple Lerch-zeta type functions,” Advanced Studies in Contemporary Mathematics, vol. 21, no. 4, pp. 367–373, 2011.

A. Bayad and T. Kim, “Identities involving values of Bernstein, -Bernoulli, and -Euler polynomials,” Russian Journal of Mathematical Physics, vol. 18, no. 2, pp. 133–143, 2011.

M. Can, M. Cenkci, V. Kurt, and Y. Simsek, “Twisted Dedekind type sums associated with Barnes' type multiple Frobenius-Euler -functions,” Advanced Studies in Contemporary Mathematics, vol. 18, no. 2, pp. 135–160, 2009.

L. Carlitz and V. E. Hoggatt, Jr., “Generalized Eulerian numbers and polynomials,” The Fibonacci Quarterly, vol. 16, no. 2, pp. 138–146, 1978.

D. Ding and J. Yang, “Some identities related to the Apostol-Euler and Apostol-Bernoulli polynomials,” Advanced Studies in Contemporary Mathematics, vol. 20, no. 1, pp. 7–21, 2010.