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 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

1  Introduction

For n0, the central factorials x[n] are given by [13]

x[0]=1,x[n]=x(x+n2-1)(x+n2-2)(x-n2+1),(n1),

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

xn=k=0nT(n,k)x[k],(n0),(see [46]).(1)

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

tet-1ext=n=0Bn(x)tnn!,(see [78]).

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

2tet-e-text=tsinhtext=n=0Dn(x)tnn!.

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 Dn(x)=2nBn(x+12),(n0). Also, we note that bn(x)=12Dn(x) is called the type 2 Bernoulli polynomials in [10]. Let n be a positive integer and let k be a nonnegative integer. As is well known, Bernoulli polynomials appear in the following expressions of the sums of powers of consecutive integers. That is

l=0n-1lk=Bk+1(n)-Bk+1(0)k+1.(2)

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

l=0n-1(2l+1)k=12(k+1)(Dk+1(2n)-Dk+1).(3)

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 p.

It is known that the Euler polynomials are given by

2et+1ext=n=0En(x)tnn!.(4)

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

Whereas the type 2 Euler polynomials are defined by

sechtext=2et+e-text=n=0En*(x)tnn!. (5)

When x = 0, En = En(0), (n0), are called the type 2 Euler numbers. We observe that En*(x)=2nEn(x+12).

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 E2n+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).

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

l=0n-1(-1)llk=Ek(n)+Ek2.

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

l=0n-1(-1)l(2l+1)k=Ek*(2n)+Ek*2.(6)

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 p.

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

1k!(et-1)k=n=kS2(n,k)tnn!,(k0),(see [2]),

and the Stirling numbers of the first kind by

1k!(log(1+t))k=n=kS1(n,k)tnn!,(see [2]).

From (6), we can derive

1k!(et2-e-t2)k=n=kT(n,k)tnn!,(k0),(7)

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

Thus, by (7), we get

T(n,k)=1k!l=0k(kl)(-1)k-l(l-k2)n,(k0).(8)

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

tlog(1+t)(1+t)x=n=0bn(x)tnn!,(see [291112]).(9)

Sometimes 1n!bn(x) are called Bernoulli polynomials of the second kind, whereas bn(x) are called Cauchy polynomials (see [2,11]). However, we will stick to our definition for the Bernoulli polynomials of the second kind.

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 [9,1315]). Here we remark that

bn=Bn(n)(1),(n0),(see (11)),

where Bn(k)(x) are the Bernoulli polynomials of order k given by

(tet-1)kext=n=0Bn(k)(x)tnn!,k,(see [1617]).

In [9], Howard studied the polynomials αn(z)(λ) given by

(λt1-(1-t)λ)z=n=0αn(z)(λ)tnn!.(10)

For any real number λ0,1, Korobov defined the degenerate Bernoulli polynomials of the second given by

λt(1+t)λ-1(1+t)x=n=0bn(x;λ)tnn!.

Then we see that limλ0bn(x;λ)=bn(x). In fact, Korobov introduced what he called ‘special polynomials’ pn(x) given by bn(x;p)=n!pn(x), for any integer p with p2 (see [18]). Here we note that bn(x;λ) are also called the Korobov polynomials of the first kind and denoted by Kn(x;λ) (see [12]).

When x = 0, bn(λ)=bn(0;λ) are called the degenerate Bernoulli numbers of the second kind. It is immediate to see that bn(λ)=(-1)nαn(1)(λ) (see (10)). Further, in [19] Howard considered the degenerate Bernoulli numbers of the second kind which is denoted by αn(λ). Note also that bn(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 order r are defined by the generating function

(tlog(1+t))r(1+t)x=n=0bn(r)(x)tnn!,r.

We note from [2123] that

bn(r)(x)=Bn(n-r+1)(x+1),(n,r0).(11)

From (9), we have

01(1+t)x+ydy=tlog(1+t)(1+t)x=n=0bn(x)tnn!,(12)

and

01(1+t)x+ydy=n=001(x+y)ndytnn!,(13)

where (x)0 = 1, (x)n=x(x-1)(x-n+1), (n1).

By (12) and (13), we get

01(x+y)ndy=bn(x),(n0).

We observe that

201(1+t)2y-1+xdy=(1+t)-(1+t)-1 log(1+t)(1+t)x.(14)

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

(1+t)-(1+t)-12log(1+t)(1+t)x=n=0bn*(x)tnn!.(15)

When x = 0, bn*=bn*(0) is called the type 2 Bernoulli numbers of the second kind.

We observe that

01(1+t)2y-1+xdy=n=001(2y-1+x)ndytnn!.(16)

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

Theorem 2.1. For n0, we have

01(2y-1+x)ndy=bn*(x).

In particular,

bn*=l=0nS1(n,l)12(l+1)(1+(-1)l),

and

bn*(1)=l=0n2lS1(n,l)1l+1.

We illustrate a few values of bn* in the following example.

Example 1: We observe first that bn*=1ln,levenS1(n,l)1l+1.

b1*=0,b2*=S1(2,2)13=13,b3*=S1(3,2)13=(-3)×13=-1,b4*=S1(4,2)13+S1(4,4)15=11×13+15=5815,b5*=S1(5,2)13+S1(5,4)15=(-50)×13+(-10)×15=-563,b6*=S1(6,2)13+S1(6,4)15+S1(6,6)17=274×13+85×15+17=11390105.

For α, let us define the type 2 Bernoulli polynomials of the second kind with order α by

((1+t)-(1+t)-12log(1+t))α(1+t)x=n=0bn*(α)(x)tnn!.(17)

When x = 0, bn*(α)=bn*(α)(0) are called the type 2 Bernoulli numbers of the second kind with order α.

From (17) and with α=k, we have

n=0bn*(k)(x)tnn!=((1+t)-(1+t)-12log(1+t))k(1+t)x.(18)

By replacing t by et2-1 in (18), we get

k!tk1k!(et2-e-t2)ket2x=l=0bl*(k)(x)1l!(et2-1)l=n=0(12nl=0nbl*(k)(x)S2(n,l))tnn!.(19)

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

k!tk1k!(et2-e-t2)ket2x=n=0(l=0n(nl)(l+kl)T(l+k,k)2-n+lxn-l)tnn!.(20)

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

Theorem 2.2. For n0 and k, we have

l=0nbl*(k)(x)S2(n,l)=l=0n(nl)(l+kl)T(l+k,k)2lxn-l.

In particular, we have

2nT(n+k,k)=(n+kn)l=0nbl*(k)S2(n,l),bn*(k)=l=0nS1(n,l)2lT(l+k,k)(l+kl).

We illustrate a few values of bn*(2) in the following example:

Example 2: Let n2 be any integer.

Then we have from (8) that T(n,2)=12!l=02(2l)(-1)2-l(l-1)n={1,ifneven,0,ifnodd.

Thus, for n1, we have bn*(2)=1ln,levenS1(n,l)2l(l+22).

b1*(2)=0,b2*(2)=S1(2,2)22(42)=46=23,b3*(2)=S1(3,2)22(42)=(-3)×46=-2,b4*(2)=S1(4,2)22(42)+S1(4,4)24(62)=11×46+1615=425,b5*(2)=S1(5,2)22(42)+S1(5,4)24(62)=(-50)×46+(-10)×1615=-44,b6*(2)=S1(6,2)22(42)+S1(6,4)24(62)+S1(6,6)26(82)=274×46+85×1615+6428=578821.

For α, we recall that the cosecant polynomials of order α are defined by

(2tet-e-t)αext=n=0Dn(α)(x)tnn!.(21)

For k, let us take α=-k and replace t by log(1+t) in (21). Then we have

((1+t)-(1+t)-12log(1+t))k(1+t)x=l=0Dl(-k)(x)1l!(log(1+t))l=n=0(l=0nS1(n,l)Dl(-k)(x))tnn!.(22)

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

Theorem 2.3. For n0, k, we have

bn*(k)(x)=l=0nS1(n,l)Dl(-k)(x).

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

1k!((1+t)-(1+t)-1)k=l=kT(l,k)2l1l!(log(1+t))l=l=kT(l,k)2ln=lS1(n,l)tnn!=n=k(l=knT(l,k)2lS1(n,l))tnn!.(23)

On the other hand, we also have

1k!((1+t)-(1+t)-1)k=1k!((1+t)-(1+t)-12log(1+t))k(2log(1+t))k=2kl=0bl*(k)tll!m=kS1(m,k)tmm!=2kn=k(m=knS1(m,k)(nm)bn-m*(k))tnn!.(24)

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

Theorem 2.4. For n, k0, we have

l=knT(l,k)2lS1(n,l)=2kl=knS1(l,k)(nl)bn-l*(k)=2kl=0n-kS1(n-l,k)(nl)bl*(k).

We observe that

0101(1+t)2(x1++xk)-k+xdx1dx2dxk=((1+t)-(1+t)-12log(1+t))k(1+t)x=n=0bn*(k)(x)tnn!.(25)

Thus, by (25), we get

1n!bn*(k)(x)=0101(2(x1++xk)-k+xn)dx1dx2dxk.

Now, for α we define the conjugate type 2 Bernoulli polynomials of the second kind with order α by

((1+t)-(1+t)-12(1+t)log(1+t))α(1+t)x=n=0b^n*(α)(x)tnn!.(26)

Then, by (26), we get

0101(1+t)-2(x1++xk)+xdx1dx2dxk=((1+t)-(1+t)-12(1+t)log(1+t))k(1+t)x=n=0b^n*(k)(x)tnn!.(27)

By (27), we get

1n!b^n*(k)(x)=0101(-2(x1++xk)+xn)dx1dx2dxk.(28)

When x = 0, b^n*(α)=b^n*(α)(0) is called the conjugate type 2 Bernoulli numbers of the second kind with order α.

For k, by (28), we get

1n!b^n*(k)(k)=0101(-2(x1++xk)+kn)dx1dx2dxk=(-1)n0101(2(x1++xk)-k+n-1n)dx1dxk=(-1)nm=0n(n-1n-m)0101(2(x1++xk)-km)dx1dxk=(-1)nm=1n(n-1m-1)1m!bm*(k).(29)

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

Theorem 2.5. For n, k, we have

(-1)n1n!b^n*(k)(k)=m=1n(n-1m-1)1m!bm*(k).

Remark. Likewise, for n, k, we have

(-1)n1n!bn*(k)(k)=m=1n(n-1m-1)1m!b^m*(k).

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.

## References

1.  Kim, T., Kim, D. S. (2020). A note on central bell numbers and polynomials. Russian Journal of Mathematical Physics, 27(1), 76–81. DOI 10.1134/S1061920820010070. [Google Scholar] [CrossRef]

2.  Roman, S. (1984). The umbral calculus, pure and applied mathematics, vol. 111. New York: Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers]. [Google Scholar]

3.  Riordan, J. (1968). Combinatorial identities. New York-London-Sydney: John Wiley & Sons, Inc. [Google Scholar]

4.  Butzer, P. L., Schmidt, M., Stark, E. L., Vogt, L. (1989). Central factorial numbers; their main properties and some applications. Numerical Functional Analysis and Optimization, 10(5–6), 419–488. DOI 10.1080/01630568908816313. [Google Scholar] [CrossRef]

5.  Howard, F. T. (1982). A theorem relating potential and bell polynomials. Discrete Mathematics, 39(2), 129–143. DOI 10.1016/0012-365X(82)90136-4. [Google Scholar] [CrossRef]

6.  Nörlund, N. E. (1924). Vorlesungen über differenzenrechnung. Berlin: Bei Springer. [Google Scholar]

7.  Bell, E. T. (1935). General relations between Bernoulli, Euler, and allied polynomials. Transactions of the American Mathematical Society, 38(3), 493–500. DOI 10.1090/S0002-9947-1935-1501824-4. [Google Scholar] [CrossRef]

8.  Kim, T., Kim, D. S. (2020). Degenerate polyexponential functions and degenerate Bell polynomials. Journal of Mathematical Analysis and Applications, 487(2), 124017. DOI 10.1016/j.jmaa.2020.124017. [Google Scholar] [CrossRef]

9.  Merlini, D., Sprugnoli, R., Verri, M. C. (2006). The Cauchy numbers. Discrete Mathematics, 306(16), 1906–1920. DOI 10.1016/j.disc.2006.03.065. [Google Scholar] [CrossRef]

10. Kim, D. S., Kim, H. Y., Kim, D., Kim, T. (2019). Identities of symmetry for type 2 Bernoulli and Euler polynomials. Symmetry, 11(5), 613. DOI 10.3390/sym11050613. [Google Scholar] [CrossRef]

11. Jordan, C. (1965). Calculus of finite differences. Bronx, New York: Chelsea. [Google Scholar]

12. Dolgy, D. V., Kim, D. S., Kim, T. (2017). On Korobov polynomials of the first kind. Russian Academy of Sciences Sbornik Mathematics, 208(1), 65–79. DOI 10.1070/SM8449. [Google Scholar] [CrossRef]

13. Blagouchine, I. V. (2018). Three notes on Ser’s and Hasse’s representations for the zeta-functions. Integers: Electronic Journal of Combinatorial Number Theory, 18A, 1–45. [Google Scholar]

14. Kowalenko, V. (2009). Generalizing the reciprocal logarithm numbers by adapting the partition method for a power series expansion. Acta Applicandae Mathematicae, 106(3), 369–420. DOI 10.1007/s10440-008-9304-5. [Google Scholar] [CrossRef]

15. Kowalenko, V. (2010). Properties and applications of the reciprocal logarithm numbers. Acta Applicandae Mathematicae, 109(2), 413–437. DOI 10.1007/s10440-008-9325-0. [Google Scholar] [CrossRef]

16. Carlitz, L. (1952). A note on Bernoulli numbers and polynomials of higher order. Proceedings of the American Mathematical Society, 3(4), 608–613. DOI 10.1090/S0002-9939-1952-0051873-6. [Google Scholar] [CrossRef]

17. Kim, D. S., Kim, T. (2020). A note on a new type of degenerate Bernoulli numbers. Russian Journal of Mathematical Physics, 27(2), 227–235. DOI 10.1134/S1061920820020090. [Google Scholar] [CrossRef]

18. Korobov, N. M. (2001). On some properties of special polynomials. Proceedings of the IV International Conference “Modern Problems of Number Theory and its Applications”, vol. 1, pp. 40–49. Chebyshevski Sbornik. [Google Scholar]

19. Howard, F. T. (1996). Explicit formulas for degenerate Bernoulli numbers. Discrete Mathematics, 162(1–3), 175–185. DOI 10.1016/0012-365X(95)00284-4. [Google Scholar] [CrossRef]

20. Araci, S. (2021). Degenerate poly-type 2-Bernoulli polynomials. Mathematical Sciences and Application E-Notes, 9(1), 1–8. [Google Scholar]

21. Adelberg, A. (1992). On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arithmetica, 62(4), 329–342. DOI 10.4064/aa-62-4-329-342. [Google Scholar] [CrossRef]

22. Adelberg, A. (1995). A finite difference approach to degenerate Bernoulli and Stirling polynomials. Discrete Mathematics, 140(1–3), 1–21. DOI 10.1016/0012-365X(93)E0188-A. [Google Scholar] [CrossRef]

23. Kim, T., Kim, D. S., Kwon, J., Lee, H. (2021). Representations of degenerate poly-Bernoulli polynomials. Journal of Inequalities and Applications, 58, 12. [Google Scholar]