On Poly Bernoulli polynomials of the second kind with umbral calculus viewpoint

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R E S E A R C H

Open Access

On Poly-Bernoulli polynomials of the second

kind with umbral calculus viewpoint

Dae San Kim

1

_{, Taekyun Kim}

2*

_{, Touﬁk Mansour}

3

_{and Dmitry V Dolgy}

4

*_{Correspondence: [email protected]} 2_{Department of Mathematics,}

Kwangwoon University, Seoul, South Korea

Full list of author information is available at the end of the article

Abstract

Poly-Bernoulli polynomials of the second kind were introduced in Kimet al.(Adv. Diﬀer. Equ. 2014:219, 2014) as a generalization of the Bernoulli polynomial of the second kind. Here we investigate those polynomials and derive further results about them by using umbral calculus.

Keywords: Bernoulli polynomials; poly-Bernoulli polynomials; Stirling numbers; umbral calculus

1 Introduction

Following Kaneko [], the poly-Bernoulli polynomials have been studied by many re-searchers in recent decades. Poly-Bernoulli polynomials B(nk)(x) were deﬁned as Lik(–e–t)

–e–t ext=

n≥B (k)

n (x)t n

n!, whereLik(x) =

r≥x

r

rk is the classical polylogarithm

func-tion, which satisﬁes _dxdLik(x) = _xLik–(x). The poly-Bernoulli polynomials have

wide-ranging applications in mathematics and applied mathematics (see [–]). Fork∈Z, the poly-Bernoulli polynomialsb(nk)(x) of the second kind are given by the generating function

Lik( –e–t) log( +t) ( +t)

x₌

n≥

b(_nk)(x)t

n

n!. (.)

Whenx= ,b(nk)=b(nk)() are called the poly-Bernoulli numbers of the second kind. When

k= ,bn(x) =b()n (x) are called the Bernoulli polynomial of the second kind (see [–]).

Poly-Bernoulli polynomials of the second kind were introduced as a generalization of the Bernoulli polynomial of the second kind (see []). The aim of this paper is to use um-bral calculus to obtain several new and interesting explicit formulas, recurrence relations and identities of poly-Bernoulli polynomials of the second kind. Umbral calculus has been used in numerous problems of mathematics. Umbral techniques have been of use in dif-ferent areas of physics; for example it is used in group theory and quantum mechanics by Biedenharnet al.(see [–]).

Letbe the algebra of polynomials in a single variablexoverCand let∗be the vector space of all linear functionals on. We denote the action of a linear functionalLon a polynomialp(x) byL|p(x). Deﬁne the vector space structure on∗bycL+cL|p(x)=

cL|p(x)+cL|p(x), wherec,c∈C(see [–]). We deﬁne the algebra of a formal power

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series in a single variabletto be

H=

f(t) =

k≥

ak

tk

k!

ak∈C

. (.)

The formal power series in the variable t deﬁnes a linear functional on by setting

f(t)|xn₌_a

n, for alln≥ (see [–]). Thus

tk|xn=n!δn,k, for alln,k≥ (see [–]), (.)

whereδn,kis the Kronecker symbol. LetfL(t) =

n≥L|xnt

n

n!. By (.), we havefL(t)|x

n₌

L|xn_{. Thus, the map}_L_→_f

L(t) is a vector space isomorphism from∗ontoH. Therefore, His thought of as a set of both formal power series and linear functionals. We callHthe

umbral algebra. Theumbral calculusis the study of the umbral algebra.

Letf(t) be a non-zero power series, theorder O(f(t)) is the smallest integerkfor which the coeﬃcient oftk _{does not vanish. If}_O₍_f₍_t_{)) =  (respectively,}_O₍_f₍_t_{)) = ), then}_f₍_t_{) is}

called adelta(respectively, aninvertable) series. Suppose thatf(t) is a delta series andg(t) is an invertable series, then there exists a unique sequencesn(x) of polynomials such that

g(t)(f(t))k_|_s

n(x)=n!δn,k, wheren,k≥. The sequencesn(x) is called theSheﬀersequence

for (g(t),f(t)) which is denoted bysn(x)∼(g(t),f(t)) (see [, ]). Forf(t)∈Handp(x)∈

, we haveeyt_|_p₍_x₎₌_p₍_y_),_f₍_t₎_g₍_t₎_|_p₍_x₎₌_g₍_t₎_|_f₍_t₎_p₍_x₎_{, and}_f₍_t_{) =}

n≥f(t)|xnt

n n! and

p(x) =_n_≥_tn_|_p₍_x₎xn

n! (see [, ]). Thus, we obtaintk|p(x)=p(k)() and|p(k)(x)=

p(k)_{(), where}_p(k)_{() denotes the}_k_{th derivative of}_p₍_x_{) with respect to}_x_at_x_{= . Therefore,}

we gettk_p₍_x_{) =}_p(k)₍_x_{) =} dk

dxkp(x), for allk≥ (see [, ]). Thus, forsn(x)∼(g(t),f(t)), we

have



g(f¯(t))e

y¯f(t)₌

n≥

sn(y)

tn

n!, (.)

for all y∈C, where ¯f(t) is the compositional inverse of f(t) (see [, ]). Forsn(x)∼

(g(t),f(t)) andrn(x)∼(h(t),(t)), letsn(x) = n

k=cn,krk(x), then we have

cn,k=



k!

h(f¯(t))

g(f¯(t))

f¯(t)kxn

(.)

(see [, ]).

It is immediate from (.) and (.) to see thatb(nk)(x) is the Sheﬀer polynomial for the

pairg(x) = t

Lik(–e–et)andf(t) =e

t_{– , that is,}

b(_nk)(x)∼

t Lik( –e–et)

,et– 

. (.)

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2 Explicit expressions

Before proceeding, we observe that

Lik

whereS(n,k) is the Stirling number of the second kind, which is deﬁned by the identity

xn₌n

which implies that



Now, we are ready to present several formulas for thenth poly-Bernoulli polynomials of the second kind.

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Thus, by (.) we have

which completes the proof.

LetS(n,k) be the Stirling number of the ﬁrst kind, which is deﬁned by the identity (x)n=

Thus, by (.) and using the arguments in the proof of Theorem ., we obtain the required

formula.

For the next explicit formula, we use the conjugation representation, namely (.).

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Ifj= , thencn,=bn(k). Thus, assume now that ≤j≤n. So

j!cn,j=

Lik

 –e–tlog( +t)j–|xn

= Lik

 –e–t(j– )!

m≥j–

S(m,j– )

tm

m!x

n

= (j– )!

n

m=j–

n m

S(m,j– )

Lik

 –e–t|xn–m,

which, by (.), implies that

j!cn,j= (j– )! n

m=j–

n m

S(m,j– )

≥

r=

(–)r+_r_! rk S(,r)

t

!

xn–m

= (j– )!

n

m=j–

n m

S(m,j– )

_n_–_m

r=

(–)r+n–mr!

rk S(n–m,r)

,

In order to state our next formula, we recall thatbn(x) =b()n (x) is the Bernoulli

poly-nomial of the second kind, which is given by the generating function _log_(+t _t₎( +t)x₌

n≥bn(x)t n n!.

Theorem . For all n≥,

b(_nk)(x) = 

n+ 

n

j=

n+ 

j

B_n(k_+–) _j–B(_nk_+–) _j(–)b(x),

where B(nk)(x)is the nth poly-Bernoulli polynomial.

Proof From the deﬁnitions, we have

b(_nk)(y) =

≥

b(k)(y) t

!

xn

= Lik( –e

–t₎ log( +t) ( +t)

y_xn

= e

–t_{– }

–t

Lik( –e–t)

 –e–t

t

log( +t)( +t)

y_xn

= e

–t_{– }

–t

Lik( –e–t)

 –e–t t

log( +t)( +t)

y_xn

= e

–t_{– }

–t

Lik( –e–t)

 –e–t

≥

b(y) t

!x

n

.

SinceB(nk)(x) is the poly-Bernoulli polynomial given by the generating functionLik(–e

–t₎

–e–t ×

ext ₌

n≥B (k)

n (x)t n

n!, we have

Lik(–e–t)

–e–t xn=B

(k)

n (x) and _dxdB(nk)(x) =nB(nk–)(x). Thusb (k)

n (y) = n

=

_n

b(y)e

–t_–

–t |B

(k)

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Proposition .. in []), we obtain

Theorem . For all n≥,

Proof By using a similar argument as in the proof of Theorem ., we obtain

b(_nk)(y) = Lik( –e

which, by (.), gives

b(_nk)(y) =

as required.

Note that the statement of Theorem . has been obtained in Theorem . of [].

3 Recurrence relations

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and

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The ﬁrst term in (.) is given by

For the second term in (.), we note that

d

which has order at least zero. So, the second term in (.) is given by

d

By substituting (.) and (.) into (.), we complete the proof.

4 Identities

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Proof We computeA=Lik( –e–t)|xn+in two diﬀerent ways. On the one hand, by (.),

By comparing (.) and (.), we obtain the required identity.

By using similar techniques as in the proof of Theorem . with computing

Lik( –e–t) log( +t)

log( +t)mxn

in two diﬀerent ways, we obtain the following result (we leave the proof as an exercise to the interested reader).

Theorem . For all n– ≥m≥,

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Theorem . For all n≥,

which gives the following identity.

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=

n–m

j=

n–j

=m

s j

n–j

S(,m)( –λ)–j(n)j

Lik( –e–t) log( +t)

xn–j–

=

n–m

j=

n–j

=m

s j

n–j

S(,m)( –λ)–j(n)jb(nk–)j–

=

n–m

j=

n–m–j

=

s j

n–j

S(n–j–,m)( –λ)–j(n)jb(k),

which gives the following identity.

Theorem . For all n≥,

b(_nk)(x) =

n

m=

_n_–_m

j=

n–m–j

=

s j

n–j

S(n–j–,m)( –λ)–j(n)jb(k)

H_m(s)(λ,x).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to this work. All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Sogang University, Seoul, South Korea.}2_{Department of Mathematics, Kwangwoon}

University, Seoul, South Korea.3_{Department of Mathematics, University of Haifa, Haifa, 31905, Israel.}4_{Institute of Natural}

Sciences, Far Eastern Federal University, Vladivostok, 690950, Russia.

Acknowledgements

This work was partially supported by Kwangwoon University in 2014.

Received: 19 November 2014 Accepted: 7 January 2015

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