Computer Modeling in Engineering & Sciences |

DOI: 10.32604/cmes.2021.016532

ARTICLE

Some Identities of the Higher-Order Type 2 Bernoulli Numbers and Polynomials of the Second Kind

1Department of Mathematics, Kwangwoon University, Seoul, 139-701, Korea

2Department of Mathematics, Sogang University, Seoul, 121-742, Korea

3Kwangwoon Glocal Education Center, Kwangwoon University, Seoul, 139-701, Korea

4Department of Mathematics Education, Gyeongsang National University, Jinju, 52828, Korea

*Corresponding Authors: Taekyun Kim. Email: kimtk2015@gmail.com; Jongkyum Kwon. Email: mathkjk26@gnu.ac.kr

Received: 17 March 2021; Accepted: 10 May 2021

Abstract: We introduce the higher-order type 2 Bernoulli numbers and polynomials of the second kind. In this paper, we investigate some identities and properties for them in connection with central factorial numbers of the second kind and the higher-order type 2 Bernoulli polynomials. We give some relations between the higher-order type 2 Bernoulli numbers of the second kind and their conjugates.

Keywords: Bernoulli polynomials of the second kind; higher-order type 2 Bernoulli polynomials of the second kind; higher-order conjugate type 2 Bernoulli polynomials of the second kind

For

and the central factorial numbers of the second kind

As is well known, the Bernoulli polynomials are defined by the generating function as

When x = 0, Bn = Bn(0) are called the Bernoulli numbers. Whereas the cosecant polynomials are defined by

When x = 0, Dn = Dn(0) are called the cosecant numbers which have been already studied in p.458 of [9]. Here we observe that

On the other hand, in [11] it is noted that

Further, in [10] we considered a random variable cooked from random variables having Laplace distributions and showed its moment is closely connected with the type 2 Bernoulli numbers [10]. Yet another thing is that we obtained some symmetric identities involving type 2 Bernoulli polynomials and power sums of consecutive odd positive integers in (3) by means of Volkenborn p-adic integrals on

It is known that the Euler polynomials are given by

When x = 0, En = En(0) are called the Euler numbers.

Whereas the type 2 Euler polynomials are defined by

When x = 0, En = En(0),

Here we would like to mention that in the literature both Euler and type 2 Euler polynomials are called Euler polynomials. Sometimes this is very confusing. Let n be a positive integer. Then, according to the definition (4), all the even Euler numbers E2n = 0. Whereas, according to the definition (5), all the odd Euler numbers

Let n be an odd positive integer. As is well known, Euler polynomials and numbers appear in the expressions of the alternating sums of powers of consecutive integers. That is

On the other hand, it is shown in [10] that

Again, in [10] we considered a random variable constructed from random variables having Laplace distributions and showed its moment is closely connected with the type 2 Euler numbers [10]. Still another thing is that we deduced certain symmetric identities involving type 2 Euler polynomials and alternating power sums of consecutive odd positive integers in (6) by using fermionic p-adic integrals on

As is well known, the Stirling numbers of the second kind are given by

and the Stirling numbers of the first kind by

From (6), we can derive

the proof of which can be found in [2].

Thus, by (7), we get

It is well known that the Bernoulli polynomials of the second kind are defined by

Sometimes

When x = 0, bn = bn(0) are variously called Bernoulli numbers of the second kind, Gregory coefficients, reciprocal logarithmic numbers, and Cauchy numbers of the first kind (see [9,13–15]). Here we remark that

where

In [9], Howard studied the polynomials

For any real number

Then we see that

When x = 0,

In the next section, we will introduce the higher-order type 2 Bernoulli numbers and polynomials of the second kind as variants of the usual higher-order Bernoulli numbers and polynomials of the second kind. We will study some properties and identities for them that are associated with central factorial numbers of the second kind and the higher-order type 2 Bernoulli polynomials. We will deduce some relations between the higher-order type 2 Bernoulli numbers of the second kind and their conjugates.

2 The Higher-Order Type 2 Bernoulli Numbers and Polynomials of the Second Kind

The Bernoulli polynomials of the second kind with order r are defined by the generating function

From (9), we have

and

where (x)0 = 1,

We observe that

Now, we define the type 2 Bernoulli polynomials of the second kind by

When x = 0,

We observe that

Therefore, by (14)–(16), we obtain the following theorem:

Theorem 2.1. For

In particular,

and

We illustrate a few values of

Example 1: We observe first that

For

When x = 0,

From (17) and with

By replacing t by

On the other hand, by making use of (7) we have

Therefore, by (19) and (20), we obtain the following theorem:

Theorem 2.2. For

In particular, we have

We illustrate a few values of

Example 2: Let

Then we have from (8) that

Thus, for

For

For

Therefore, by (18) and (22), we obtain the following theorem:

Theorem 2.3. For

Replacing t by

On the other hand, we also have

Therefore, by (23) and (24), we obtain the following theorem:

Theorem 2.4. For n,

We observe that

Thus, by (25), we get

Now, for

Then, by (26), we get

By (27), we get

When x = 0,

For

Therefore, by (29), we obtain the following theorem:

Theorem 2.5. For n,

Remark. Likewise, for n,

In Section 2, we introduced the higher-order type 2 Bernoulli numbers and polynomials of the second kind and the higher-order conjugate type 2 Bernoulli numbers of the second kind. In Theorems 2–4, we obtained some properties and identities for them that are associated with central factorial numbers of the second kind and higher-order cosecant polynomials and the Stirling numbers of the first kind. In Theorem 5, we derived the relation between the higher-order type 2 Bernoulli numbers of the second kind and their conjugates.

Many problems in science and engineering can be modeled by polynomial optimization which concerns optimizing a polynomial subject to polynomial equations and inequalities. Thanks to an adoption of tools from real algebraic geometry, semidefinite programming and the theory of moments, etc., there has been tremendous progress in this field. We hope that the polynomials newly introduced in the present paper or their possible multivariate versions will play some role in near future.

Acknowledgement: The authors thank to Jangjeon Institute for Mathematical Science for the support of this research.

Funding Statement: This work was supported by the National Research Foundation of Korea (NRF) Grant Funded by the Korea Government (No. 2020R1F1A1A01071564).

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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