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

Taekyun Kim,Dae San Kim,Dmitry V.Dolgy,Si-Hyeon Lee and Jongkyum Kwon

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

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

1 Introduction

For n ≥0, the central factorials x[n]are given by [1-3]

and the central factorial numbers of the second kind T(n,k) by

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

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

Whenx=0,Dn=Dn(0)are called the cosecant numbers which have been already studied in p.458 of [9].Here we observe thatAlso, we note thatbn(x)=is called the type 2 Bernoulli polynomials in [10].Letnbe a positive integer and letkbe a nonnegative integer.As is well known, Bernoulli polynomials appear in the following expressions of the sums of powers of consecutive integers.That is

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 Volkenbornp-adic integrals on Zp.

It is known that the Euler polynomials are given by

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

Whereas the type 2 Euler polynomials are defined by

Whenx=0,En=En(0),(n≥0), are called the type 2 Euler numbers.We observe thatE*n(x)=

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.Letnbe a positive integer.Then,according to the definition (4), all the even Euler numbersE2n=0.Whereas, according to the definition (5), all the odd Euler numbersE*2n+1=0.To avoid a possible confusion, we call the polynomials in (5) the type 2 Euler polynomials, while reserving the term Euler polynomials for the ones in (4).

Letnbe 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 fermionicp-adic integrals on Zp.

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

Whenx=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

whereB(k)n (x)are the Bernoulli polynomials of orderkgiven by

In [9], Howard studied the polynomialsα(z)n (λ)given by

For any real numberλ／=0,1, Korobov defined the degenerate Bernoulli polynomials of the second given by

Then we see that limλ→0bn(x;λ)=bn(x).In fact, Korobov introduced what he called ‘special polynomials’pn(x)given bybn(x;p)=n!pn(x), for any integerpwithp≥2 (see [18]).Here we note thatbn(x;λ)are also called the Korobov polynomials of the first kind and denoted byKn(x;λ)(see [12]).

Whenx=0,bn(λ)=bn(0;λ)are called the degenerate Bernoulli numbers of the second kind.It is immediate to see thatbn(λ)=(-1)nα(1)n (λ)(see (10)).Further, in [19] Howard considered the degenerate Bernoulli numbers of the second kind which is denoted byαn(λ).Note also thatbn(λ)=Kn(0;λ).In light of these considerations,bn(λ)may be variously called the degenerate Bernoulli numbers of the second, Howard numbers and Korobov numbers of the first kind (see[20]).

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 orderrare defined by the generating function

We note from [21-23] that

From (9), we have

and

where(x)0=1,(x)n=x(x-1)...(x-n+1),(n≥1).

By (12) and (13), we get

We observe that

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

Whenx=0,b*n=b*n(0)is called the type 2 Bernoulli numbers of the second kind.

We observe that

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

Theorem 2.1.Forn≥0, we have

In particular,

and

We illustrate a few values ofb*nin the following example.

Example 1:We observe first that

Forα∈R, let us define the type 2 Bernoulli polynomials of the second kind with orderαby

Whenx=0,are called the type 2 Bernoulli numbers of the second kind with orderα.

From (17) and withα=k∈N, we have

By replacingtbyin (18), we get

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

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

Theorem 2.2.Forn≥0 andk∈N, we have

In particular, we have

We illustrate a few values ofb*(2)nin the following example:

Example 2:Letn≥2 be any integer.

Then we have from (8) thatT(n,2)=

Thus, forn≥1, we have

Forα∈R, we recall that the cosecant polynomials of orderαare defined by

Fork∈N, let us takeα=-kand replacetby log(1+t)in (21).Then we have

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

Theorem 2.3.Forn≥0,k∈N, we have

Replacingtby 2log(1+t)in (7), we derive the following equation:

On the other hand, we also have

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

Theorem 2.4.Forn,k≥0, we have

We observe that

Thus, by (25), we get

Now, forα∈R we define the conjugate type 2 Bernoulli polynomials of the second kind with orderαby

Then, by (26), we get

By (27), we get

Whenx=0,(0)is called the conjugate type 2 Bernoulli numbers of the second kind with orderα.

Fork∈N, by (28), we get

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

Theorem 2.5.Forn,k∈N, we have

Remark.Likewise, forn,k∈N, we have

3 Conclusions

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.