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 [118]). 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 [618]). 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 [15, 1924]). Here , then is called the th Euler number. From (1.6), we have (see [35, 1923]).

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 [13, 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 [618]).

Let be the space of continuous functions on . For , the fermionic -adic -integral on is defined by (see [7, 1013]). 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 [513]).

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.