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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