DIFFERENTIAL EQUATIONS ASSOCIATED WITH TANGENT NUMBERS

Abstract

In this paper, we study differential equations arising from the generating functions of tangent numbers. We give explicit identities for the tangent numbers.

1. Introduction

Recently, many mathematicians have studied in the area of the Bernoulli numbers, Euler numbers, Genocchi numbers, and tangent numbers (see [2,3,4,6,7,8]). Tangent numbers Tn and polynomials, Tn(x)(n ≥ 0), were introduced by Ryoo (see [5]). The tangent numbers Tn are defined by the generating function:

We introduce the tangent polynomials Tn(x) as follows:

In [6], Tangent numbers of higher order, are defined by means of the following generating function

The first few of them are

Nonlinear differential equations arising from the generating functions of special polynomials are studied by T. Kim and D. Kim in order to give explicit identities for special polynomials(see [1,4]). In this paper, we study differential equations arising from the generating functions of tangent numbers. We give explicit identities for the tangent numbers.

 

2. Differential equations associated with tangent numbers

In this section, we study linear differential equations arising from the generating functions of tangent numbers. Let

Then, by (2.1), we have

and

Continuing this process, we can guess that

Taking the derivative with respect to t in (2.4), we have

On the other hand, by replacing N by N + 1 in (2.4), we get

By (2.5) and (2.6), we have

Comparing the coefficients on both sides of (2.7), we obtain

and

In addition, by (2.4), we get

Thus, by (2.10), we obtain

It is not difficult to show that

Thus, by (2.12), we also get

From (2.8), we note that

and

For i = 2, 3, 4 in (2.9), we have

and

Continuing this process, we can deduce that, for 2 ≤ i ≤ N + 1,

Here, we note that the matrix ai(j)1≤i≤N+2,0≤j≤N+1 is given by

Now, we give explicit expressions for ai(N + 1). By (2.14) and (2.15), we get

and

Continuing this process, we have

Therefore, by (2.16), we obtain the following theorem.

Theorem 2.1. For N = 0, 1, 2, . . . , the functional equations

have a solution

where

Here is a plot of the surface for this solution. In Figure 1, we plot of the shape

FIGURE 1.The shape for the solution F(t)

for this solution.

From (1.1), we note that

From Theorem 1, (1.3), and (2.17), we can derive the following equation:

By comparing the coefficients on both sides of (2.18), we obtain the following theorem.

Theorem 2.2. For k,N = 0, 1, 2, . . . , we have

Let us take k = 0 in (2.19). Then, we have the following corollary.

Corollary 2.3. For N = 0, 1, 2, . . . , we have