IDENTITIES INVOLVING TANGENT NUMBERS AND POLYNOMIALS

Abstract

In this paper we give some properties, explicit formulas, several identities, a connection with tangent numbers and polynomials, and some integral formulas.

1. Introduction

Recently, many mathematicians have studied in the area of the Bernoulli numbers, Euler numbers, Genocchi numbers, and tangent numbers (see [1,2,3,4,5,8,9 10,11]). Throughout this paper, we always make use of the following notations: ℕ denotes the set of natural numbers and ℤ+ = ℕ ∪ {0} , and ℂ denotes the set of complex numbers. The tangent numbers Tn are defined by the generating function:

where we use the technique method notation by replacing Tn by Tn(n ≥ 0) symbolically [6,7]. We consider the tangent polynomials Tn(x) as follows:

Note that Numerous properties of tangent number are known. More studies and results in this subject we may see references [6,7,8]. About extensions for the tangent numbers can be found in [6,7,8]. Because

it follows the important relation

Since

we see that

Since Tn(0) = Tn, by (1.2), we have the following theorem.

Theorem 1.1. For n ∈ ℕ, we have

From (1.1), we can derive the following equation:

By comparing the coefficients on both sides of (1.3), we have the following theorem.

Theorem 1.2. For any positive integer n, we have

Now we observed that

By (1.5), we have the following theorem.

Theorem 1.3. For any positive integer n, we have

The beta integral is defined for Re(x) > 0, Re(y) > 0 by

For Re(x) > 0, the gamma function Γ(x) is defined by

The above integral for Γ(x) is sometimes called the Eulerian integral of the second kind. Thus, by (1.7) and (1.8), we have

Our aim in this paper is to give some properties, explicit formulas, several identities, a connection with tangent numbers and polynomials, and some integral formulas.

 

2. Identities involving tangent numbers and polynomials

In this section, we obtain several new and interesting identities involving tangent numbers and polynomials.

By (1.6), we get

Since

we have the following theorem.

Theorem 2.1. For n ∈ ℤ+, we have

By (2.2), we note that

From (1.4) and (2.3), we note that

Therefore, by (2.3) and (2.4), we obtain the following theorem.

Theorem 2.2. For n ∈ ℤ+, we have

For n ∈ ℕ with n ≥ 4, we obtain

Continuing this process, we obtain

Hence, by (2.3) and (2.5), we have the following theorem.

Theorem 2.3. For n ∈ ℕ with n ≥ 2, we have

By Theorem 2.2 and Theorem 2.3, we have the following corollary.

Corollary 2.4. For n ∈ ℕ with n ≥ 2, we have

From (1.4), we have (-1)nTn = Tn(2). Putting x = 1 in Theorem 2.3 gives the identity

Hence we have the following corollary.

Corollary 2.5. For n ∈ ℕ with n ≥ 2, we have

Putting x = 1 in Corollary 2.4 yields an identity

Now we observe that

For m, n ∈ ℕ with m, n ≥ 2, we have

Continuing this process, we get

By (2.6) and (2.7), we have the following theorem.

Theorem 2.6. For m, n ∈ ℕ, we have