Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

1 Issue per year


IMPACT FACTOR 2016 (Open Mathematics): 0.682
IMPACT FACTOR 2016 (Central European Journal of Mathematics): 0.489

CiteScore 2016: 0.62

SCImago Journal Rank (SJR) 2016: 0.454
Source Normalized Impact per Paper (SNIP) 2016: 0.850

Mathematical Citation Quotient (MCQ) 2016: 0.23

Open Access
Online
ISSN
2391-5455
See all formats and pricing
More options …
Volume 14, Issue 1 (Jan 2016)

Issues

Fully degenerate poly-Bernoulli numbers and polynomials

Taekyun Kim
  • Corresponding author
  • Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300387, China and Department of Mathematics, Kwangwoon University, Seoul 139-701, Korea (Republic of)
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Dae San Kim / Jong-Jin Seo
Published Online: 2016-07-26 | DOI: https://doi.org/10.1515/math-2016-0048

Abstract

In this paper, we introduce the new fully degenerate poly-Bernoulli numbers and polynomials and inverstigate some properties of these polynomials and numbers. From our properties, we derive some identities for the fully degenerate poly-Bernoulli numbers and polynomials.

Keywords: Fully degenerate poly-Bernoulli polynomial; Fully degenerate poly-Bernoulli number; Umbral calculus

MSC 2010: 11B75; 11B83; 05A19; 05A40

1 Introduction

It is well known that the Bernoulli polynomials are defined by the generating function

tetext=n=0Bn(x)tnn!,(see[121]).(1)

When x = 0, Bn = Bn (0) are called the Bernoulli numbers. From (1), we note that

Bn(x)=i=0nnlBlxn1,(n0),(2)

and

B0=1,Bn(1)Bn=δ1,n,(nN),(see[1,19]),(3)

where δn k is the Kronecker’s symbol.

In [3], L. Carlitz considered the degenerate Bernoulli polynomials which are given by the generating function

t(1+λt)1λ1(1+λt)xλ=n=0βn,λ(x)tnn!.(4)

When x = 0, βn, λ = βn, λ (0) are called the degenerate Bernoulli numbers. From (1) and (4), we note that

tet1ext=limλ0t(1+λt)1λ1(1+λt)xλ=n=0limλ0βn,λ(x)tnn!(5)

Thus, by (5), we get

Bn(x)=limλ0βn,λ(x),(see[3,15]).(6)

By (4), we get

n=0(βm,λ(1)βm,λ)tnn!=t,(7)

and

βn,λ(x)=l=0nnlβl,λλn1xλn1.(8)

From (7), we have

βn,λ1βn,λ=δ1,n,(n0),β0,λ=1.(9)

Now, we consider the degenerate Bernoulli polynomials which are different from the degenerate Bernoulli polynomials of L. Carlitz as follows:

log1+λt1λ1+λt1λ11+λtxλ=n=0bn,λ(x)tnn!.(10)

When x = 0, bn, λ = bn, λ (0) are called the degenerate Bernoulli numbers.

Note. The degenerate Bernoulli polynomials are also called Daehee polynomials with λ-parameter (see [13]).

From (10), we note that

tet1ext=limλ0log1+λt1λ1+λt1λ1+λtxλ=n=0limλ0bn,λ(x)tnn!.(11)

By (1) and (11), we see that

Bn(x)=limλ0bn,λ(x),(n0).

The classical polylogarithm function Lik (x) is defined by

Lik(t)=n=1tnnk,kZ,(see[10,11]).(12)

It is known that the poly-Bernoulli polynomials are defined by the generating function

Lik1et1etext=n=0Bn(k)(x)tnn!,(see[9,10,12]).(13)

When k = 1, we have

n=0Bn(1)(x)tnn!=t1etext=tet1e(x+1)t=n=0Bn(x+1)tnn!.(14)

Bn(1)(x)=Bn(x+1),n0.

By (14), we easily get

Let x = 0. Then Bn(k)=Bn(k)(0) are called the poly-Bernoulli numbers.

In this paper, we introduce the new fully degenerate poly-Bernoulli numbers and polynomials and inverstigate some properties of these polynomials and numbers. From our investigation, we derive some identities for the fully degenerate poly-Bernoulli numbers and polynomials.

2 Fully degenerate poly-Bernoulli polynomials

For k ∈ ℤ, we define the fully degenerate poly-Bernoulli polynomials which are given by the generating function

Lik11+λt1λ11+λt1λ1+λtxλ=n=0βn,λ(k)(x)tnn!.(15)

When x=0,βn,λ(k)=βn,λ(k)(0) are called the fully degenerate poly-Bernoulli numbers. From (13) and (15), we have

Lik1et1etext=limλ0Lik11+λt1λ11+λt1λ1+λtxλ=n=0limλ0βn,λ(k)(x)tnn!.(16)

Thus, we get

limλ0βn,λ(k)(x)=Bn(k)(x),n0.(17)

By (15), we get

n=0βn,λ(k)(x)tnn!=Lik11+λt1λ11+λt1λ1+λtxλ=n=0l=0nnlβl,λ(k)xλn1λn1tnn!.(18)

Thus, from (18), we have

βn,λ(k)x+y=l=0nnlyλn1λn1βl,λ(k)(x),n0,(19)

and

βn,λ(k)(x)=l=0nnlxλn1λn1βl,λ(k).

Therefore, by (17) and (19), we obtain the following theorem.

For k ∈ ℤ, ≥ 0, we have

βn,λ(k)(x+y)=l=0nnlyλn1λn1βl,λ(k)(x),n0,(20)

and

limλ0βn,λ(k)(x)=Bn(k)(x),

where (x)n=x(x1)(xn+1)=l=0nS1(n,l)xl..

From (15), we can derive the following equation:

n=0βn,λ(k)(x)tnn!=Lik11+λt1λ1+λt1λ11+λtx+1λ.(21)

Thus, by (21), we get

n=0βn,λ(k)βn,λ(k)(1)tnn!=Lik1(1+λt)1λ=m=11(1+λt)1λmmk=m=0(1)m+1(m+1)ke1λlog(1+λt)1m+1=m=0(1)m+1(m+1)k(m+1)!l=m+1XS2(l,m+1)(1)lλl(log(1+λt))ll!=l=1m=0l1(1)m+1(m+1)k(m+1)!S2(l,m+1)(1)lλln=lS1(n,l)λntnn!=n=1l=1nm=0l1m!(m+1)(1)lm1λnlS2(l,m+1)S1(n,l)(m+1)ktnn!,(22)

where S2 (n, l) and S1 (n, l) are the Stirling numbers of the second kind and of the first kind, respectively. Therefore, by (22), we obtain the following theorem.

For k ∈ ℤ, n ≥ 1, we have

βn,λ(k)βn,λ(k)(1)=l=1nm=0l1m!(1)lm1λn1S2l,m+1S1n,l(m+1)k1.

From (12), we can easily derive the following equation:

Lik(t)=ddtLik(t)=1tLik1(t).(23)

Thus, by (23), the generating function of the fully degenerate poly-Bernoulli numbers is also written in terms of the following iterated integral:

(1+λt)1λ(1+λt)1λ10t1(1+λt)1λ1(1+λt)×0t1(1+λt)1λ1(1+λt)0tlog(1+λt)1λ(1+λt)1λ1(1+λt)dtdtk1times=n=0βn,λ(k)tnn!.(24)

For k = 2, we have

n=0βn,λ(2)tnn!=(1+λt)1λ(1+λt)1λ10tlog(1+λt)1λ(l+λt)1λ1(1+λt)λλdt=(1+λt)1λ(1+λt)1λ1m=0bm,λ(λ)1m!0ttmdt=t(1+λt)1λ1(1+λt)1λm=0bm,λ(λ)(m+1)tmm!=n=0l=0nnlβl,λ(1)bnl,λ(λ)nl+1tnn!.(25)

Therefore, by (25), we obtain the following theorem.

For n ≥ 0, we have

βn,λ(2)=l=0nnlβl,λ(1)bn1,λλnl+1.

Note that

Bn2=limλ0βn,λ(2)=l=0nnlBl(1)Bn1nl+1.

From (15), we have

n=0βn,λ(k)tnn!=Lik1(1+λt)1λ1(1+λt)1λ=m=01(m+1)k1(1+λt)1λm=m=0(1)m(m+1)ke1λlog(1+λt)1m=m=0(1)m(m+1)km!l=mS2(l,m)1λl(log(1+λt))ll!=l=0m=0l(1)m+lm!(m+1)kS2(l,m)λl1l!(log(1+λt))l=l=0m=0l(1)m+lm!(m+1)kS2(l,m)λln=lS1(n,l)λntnn!=n=0l=0nm=0l(1)m+lm!(m+1)kS2(l,m)S1(n,l)λnltnn!.(26)

Therefore, by (26), we obtain the following theorem.

For n ≥ 0, we have

βn,λ(k)=l=0nm=0l1m+lm!m+1kS2l,mS1n,lλnl.

Note that

Bn(k)=limλ0βn,λ(k)=m=0n1m+nm!m+1kS2n,m.

From (23), we have

ddtLik11+λt1λ=111+λt1λ1+λt1λ1Lik111+λt1λ=1+λt1λ1n=0βn,λ(k1)tnn!.(27)

On the other hand,

ddtLik1(1+λt)1λ=ddt1(1+λt)1λ11(1+λt)1λLik1(1+λt)1λ)=(1+λt)1λ111(1+λt)1λLik1(1+λt)1λ+1(1+λt)1λddtn=0βn,λ(k)tnn!=(1+λt)1λ1n=0βn,λ(k)tnn!+1(1+λt)1λn=1βn,λ(k)tn1(n1)!.(28)

By (27) and (28), we get

n=0βn,λ(k1)tnn!=n=0βn,λ(k)tnn!+(1+λt)(1+λt)1λ1n=1βn,λ(k)tn1(n1)!=n=0βn,λ(k)tnn!+(1+λt)1λ1m=0βm+1,λ(k)tmm!+λ(1+λt)1λ1m=0βm,λ(k)mtmm!=n=0βn,λ(k)tnn!l=11λlλttll!m=0βm+1,λ(k)tmm!+λl=11λlλttll!m=0βm,λ(k)mtmm!=n=0βn,λ(k)tnn!+n=1m=0n11λnmλnmn!(nm)!m!βm+1,λ(k)tnn!+λn=1m=0n11λnmλnmmn!(nm)!m!βm,λ(k)tnn!=n=0βn,λ(k)tnn!+n=1m=0n1nm1λnmλnmβm+1,λ(k)tnn!+λn=1m=0n1nm1λnmλnmmβm,λ(k)tnn!,(29)

where

(1λ)n=1λ1λ11λn+1=l=0nS1nn,lλl,n0.

Thus, by (29), we have

βn,λk1)=βn,λk)+m=0n1nm(1λ)nmλnmβm+1,λk+λm=0n1nm(1λ)nmλnmmβm,λk=n+1βn,λk)+m=1n1nm1(1λ)nm+1λnm+1βm,λk+λm=0n1nm(1λ)nmλnmmβm;λ,kn1.(30)

Therefore, by (30), we obtain the following theorem

For n ≥ 1, we have

βn,λk)=1n+1βn,λk1)m=1n1nm1βm,λk)(1λ)nm+1λnm+1λm=0n1nm(1λ)nmλnmmβm;λk.

Note that

Bn(k)=limλ0βn,λ(k)=1n+1Bn(k1)m=1n1nm1Bm(k).

Now, we observe that

n=0(1(1+λt)1λ)n(n+1)k=n=1(1(1+λt)1λ)nnk11(1+λt)1λ=11(1+λt)1λLik(11+λt)1λ=n=0Xβn,λk)tnn!.(31)

By (31), we get

k=0n=0βn,λ(k)xnn!ykk!=k=0m=0(1(1+λt)1λ)m(m+1)kykk!=m=01(1+λx)1λmk=0(m+1)kykk!=m=0(1(1+λx)1λ)mem+1)y=j=0(1)j(e1λlog(1+λx)1)jej+1)y=j=0(1)jj!m=jS2m,j(1)mλm(log1+λx)mm!e(j+1)y=m=0j=0m(1)j+mj!S2m,jλmn=mS1n,mλnxnn!e(j+1)y=n=0m=0nj=0m(1)j+mj!S2m;jλnmS1n,mej+1)yxnn!=k=0n=0m=0nj=0m(1)j+mj!λnmS2m,jS1n,m(j+1)kxnn!ykk!.(32)

Therefore, by (32), we obtain the following theorem.

For k ∈ ℤ and n ≥ 0, we have

βn,λ(k)=m=0nj=0m(1)j+mj!λnm(j+1)kS2m,jS1n,m.

Note that

Bn(k)=limλ0βn,λ(k)=j=0n(1)j+nj!(j+1)ks2(n,j).

From Theorem 2.1, we have

ddxβn,λ(k)(x)=l=0nlnβl,λ(k)ddxi=0nl1(xiλ)=l=0nlnβl,λ(k)j=0nl11(xjλ)i=0nl1(xiλ).

The generalized falling factorial (x | λ)n is given by

(x|λ)n=x(xλ)(x2λ)(x(n1)λ),(n0).(33)

As is well known, the Bernoulli numbers of the second kind are defined by the generating funcdtion

tlog(1+t)=n=0bntnn!,(see[20]).(34)

We observe that

01(1+λt)xλdx=n=0λn01xλndxtnn!=n=001x|λndxtnn!.(35)

On the other hand,

01(1+λt)xλdx=λlog(1+λt)(1+λt)1λ1=λtlog1+λt(1+λt)1λ1t=m=0bmλmtmm!l=0(1|λ)l+1l+1tll!=n=0l=0n(1|λ)l+1l+1λnlbnlnltnn!.(36)

From (35) and (36), we have

01(x|λ)ndx=l=0nnlλnlbn1(1|λ)l+1l+1,n0.(37)

By Theorem 2.1, we get

01βn,λ(k)xdx=l=0nnlβl,λ(k)01xλnlλnldx=l=0nnlβnl,λ(k)01(x|λ)ldx=l=0nm=0llmλlmblm(1|λ)m+1m+1nlβnl,λ(k).

3 Further remarks

Let ℂ be complex number field and let F be the set of all formal power series in the variable t over ℂ with

F=f(t)=k=oaktkk!akC.(38)

Let ℙ be the algebra of polynomials in a single variable x over ℂ and let ℙ* be the vector space of all linear functionals on ℙ. The action of linear functional L ∈ ℙ* on a polynomial p (x) is denoted by 〈L| p (x)〉, and linearly extended as

cL+cL|p(x)=cL|p(x)+cL|p(x),

where c, c′ ∈ ℂ.

For f(t)=k=0aktkk!, we define a linear functional on ℙ by setting

f(t)|xn=anforalln0.(39)

Thus, by (39), we get

tkxn=n!δn,k,(n,k0),(see[4,16,20]).(40)

For fL(t)=k=0L|xktkk!, by (40), we get 〈fL (i)| xn〉 = 〈L| xn〉. In addition, the mapping LfL (t) is a vector space isomorphism from ℙ* onto F . Henceforth, F denotes both the algebra of the formal power series in t and the vector space of all linear functionals on ℙ and so an element f (i) of F can be regarded as both a formal power series and a linear functional. We refer to F umbral algebra. The umbral calculus is the study of umbral algebra (see [5, 15, 20]). The order o (f (i)) of the non-zero power series f (i) is the smallest integer k for which the coefficient of tk does not vanish.

If o (f (t)) = 1(respectively, o (f (t)) = 0), then f (t) is called a delta (respectively, an invertible) series (see [20]). For o (f (t)) = 1 and o (g (t)) = 0, there exists a unique sequence sn (x) of polynomials such that 〈g(t) f (t)k| sn (x〉 = n!δn, k, (n, k ≥ 0).

The sequence sn (x) is called the Sheffer sequence for (g (t), f (t)), and we write sn (x) ~ (g (t), f (t)) (see [20]).

Let f (t) ∈ F and p (x) ∈ ℙ. Then, by (40), we get

f(t)=k=of(t)xktkk!,p(x)=k=otkp(x)xkk!.(41)

From (41), we have

p(k)(0)=tkp(x)=1p(k)(x),(42)

where pk(x)=dkdxkp(x), (see [11, 14, 20]).

By (42), we easily get

tkp(x)=pk(x),eytp(x)=p(x+y),andeytp(x)=p(y).(43)

From (43), we have

eyt1tp(x)=xx+yp(u)du,eyt1p(x)=p(y)p(0).

Let f (t) be the linear functional such that

f(t)p(x)=oyp(u)du,(44)

for all polynomials p (x). Then it can be determined by (41) to be

f(t)=k=0f(t)|xkk!tk=k=0yk+1(k+1)!tk=1t(eyt1).(45)

Thus, for p (x) ∈ ℙ, we have

eyt1tp(x)=0yp(u)du.(46)

It is known that

sn(x)(g(t),f(t))1gf¯(t)exf¯(t)=k=0sk(x)tkk!,(xC)(47)

where f¯(t) is the compositional inverse of f (t) such that ff¯(t)=f¯(f(t))=t (see [11, 20]). From (15), we note that

βn,λ(k)(x)1etLik(1et),1λeλt1.(48)

That is,

n=0βn,λ(k)(x)tnn!=Lik1(1+λt)1λ1(1+λt)1λ(1+λt)xλ.

Thus, by (48),

1λ(eλt1)βn,λ(k)(x)=nβn1,λ(k)(x).(49)

On the other hand,

(eλt1)βn,λ(k)(x)=βn,λ(k)(x+λ)βn,λ(k)(x).(50)

Therefore, by (49) and (50), we obtain the following theorem.

For n ∈ ℕ, we have

λβn1,λ(k)(x)=1nβn,λ(k)(x+λ)βn,λ(k)(x).

By (46), we get

eyt1tβn,λ(k)(x)=xx+yβn,λ(k)(u)du.(51)

From (51), we have

eyt1tβn,λ(k)(x)=0yβn,λ(k)(u)du.(52)

Thus, by (52), we get

et1tβn,λ(k)(x)=01βn,λ(k)(u)du=l=0nm=0llmnlλlmblmβnl,λk(1|λ)m+1m+1.(53)

Therefore, by (53), we obtain the following theorem.,

For n ≥ 0 we have

et1tβn,λ(k)(x)=l=0nm=0llmnlλlmblmβnl,λk(1|λ)m+1m+1.

Note that

et1tBn(k)(x)=limλ0et1tβn,λ(k)(x)=limλ001βn,λ(k)(u)du=l=0nnlBnlk)1l+1

Let

Pn={pxCx|degpxn},n0.

For p(x) ∈ ℙn with p(x)=m=0namβm,λ(k)(x), we have

1etLik1et1λeλt1mpx=l=0nal1etLik1et1λeλt1mβl,λk)x(54)

From (48), we note that

1etLik1et1λeλt1mβl,λk)x=l!δl,m.(55)

By (54) and (55), we get

am=1m!1etLik1et1λeλt1mpx,(m0).(56)

Therefore, by (56), we obtain the following theorem.

For p (x) ∈ ℙn, we have

px=m=0namβm,λ(k)(x),

where

am=1m!1etLik1et1λeλt1mpx.

For example, let us take p(x)=βn(k)(x)(n0). Then, by Theorem 3.3, we have

Bn(k)x=m=0namβm,λ(k)(x),(57)

where

am=1m!1etLik1et1λeλt1mBn(k)x=1m!1etLik1et1λeλt1mLik1et1etxn=λmm!eλt1mxn=λml=mS2l,mλll!tlxn=λnmS2n,m.(58)

From (57) and (58), we have

Bn(k)x=m=0nλnmS2n,mβm,λ(k)x.(59)

References

  • [1]

    Araci S., Acikgoz M., Kilicman A., Extended p-adic q-invariant integrals on ℤp associated with applications of umbral calculus, Adv. Difference Equ., 2013, 2013:96, 14 pp.Google Scholar

  • [2]

    Bayad A., Simsek Y., Srivastava H. M., Some array type polynomials associated with special numbers and polynomials, Appl. Math. Comput., 2014, 244, 149–157.Web of ScienceGoogle Scholar

  • [3]

    Carlitz L., Degenerate Stirling, Bernoulli and Eulerian numbers, Utilitas Math., 1979, 15, 51–88. Google Scholar

  • [4]

    Dere R., Simsek Y., Applications of umbral algebra to some special polynomials, Adv. Stud. Contemp. Math. (Kyungshang), 2012, 22, 433–438.Google Scholar

  • [5]

    Dere R., Simsek Y., Hermite base Bernoulli type polynomials on the umbral algebra, Russ. J. Math. Phys. 2015, 22, 1–5. Google Scholar

  • [6]

    Ding D., Yang J., Some identities related to the Apostol-Euler and Apostol-Bernoulli polynomials, Adv. Stud. Contemp. Math. (Kyungshang), 2010, 20, 7–21.Google Scholar

  • [7]

    Gaboury S., Tremblay R., and B.-J. Fugere, Some explicit formulas for certain new classes of Bernoulli, Euler and Genocchi polynomials, Proc. Jangjeon Math. Soc., 2014, 17, 97–104.Google Scholar

  • [8]

    He Y., Zhang W., A convolution formula for Bernoulli polynomials, Ars Combin., 2013, 108, 97–104.Google Scholar

  • [9]

    Kaneko M., Masanobu poly-Bernoulli numbers, J Théor. Nombres Bordeaux, 1997, 9, 221–228.Google Scholar

  • [10]

    Kim D., Kim T., A note on poly-Bernoulli and higher-order poly-Bernoulli polynomials, Russ. J. Math. Phys. 2015, 22, 26–33.Google Scholar

  • [11]

    Kim D. S., Kim T., Higher-order Cauchy of the first kind and poly-Cauchy of the first kind mixed type polynomials, Adv. Stud. Contemp. Math. (Kyungshang), 2013, 23, 621–636. Google Scholar

  • [12]

    Kim D. S., Kim T., Higher-order Frobenius-Euler and poly-Bernoulli mixed-type polynomials, Adv. Difference Equ., 2013, 2013:251, 13 pp.Google Scholar

  • [13]

    Kim D. S., Kim T., Daehee polynomials with q-parameter, Adv. Studies Theor. Phys., 2014, 8, 561–569. Google Scholar

  • [14]

    Kim D. S., Kim T., Mansour T., and Dolgy D. V., On poly-Bernoulli polynomials of the second kind with umbral calculus viewpoint, Adv. Difference Equ., 2015, 2015:27, 13 pp. Google Scholar

  • [15]

    Kim T., Barnes’ type multiple degenerate Bernoulli and Euler polynomials, Appl. Math. Comput. 2015, 258, 556–564. Google Scholar

  • [16]

    Kim T. and Mansour T., Umbral calculus associated with Frobenius-type Eulerian polynomials, Russ. J. Math. Phys., 2014, 21, 484–493.Google Scholar

  • [17]

    Luo Q.-M., Some recursion formula and relations for Bernoulli numbers and Euler numbers of higher order, Adv. Stud. Contemp. Math. (Kyungshang), 2005, 10, 63–70.Google Scholar

  • [18]

    Ozden H., Simsek Y., Srivastava H. M., A unified presentation of the generating functions of the generalized Bernoulli, Euler and Genocchi polynomials, Comput. Math. Appl., 2010, 60, 2779–2787.Google Scholar

  • [19]

    Qi F., An integral representation, complete monotonicity, and inequalities of Cauchy numbers of the second kind, J. Number Theory, 2014, 144, 244–255.Google Scholar

  • [20]

    Roman S., The umbral calculus, Pure and Applied Mathematics, vol. 111, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1984. Google Scholar

  • [21]

    Sen E., Theokind on Apostol-Euler polynomials of higher order arising from Euler basis, Adv. Stud. Contemp. Math. (Kyungshang), 2013, 23, 337–345.Google Scholar

About the article

Received: 2015-05-26

Accepted: 2016-06-29

Published Online: 2016-07-26

Published in Print: 2016-01-01


Citation Information: Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2016-0048.

Export Citation

© 2016 Kim et al., published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Comments (0)

Please log in or register to comment.
Log in