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R E S E A R C H

Open Access

Barnes’ multiple Bernoulli and generalized

Barnes’ multiple Frobenius-Euler mixed-type

polynomials

Dae San Kim

1

, Taekyun Kim

2*

, Takao Komatsu

3

and Hyuck In Kwon

2

*Correspondence: tkkim@kw.ac.kr 2Department of Mathematics,

Kwangwoon University, Seoul, 139-701, Republic of Korea Full list of author information is available at the end of the article

Abstract

In this paper, by considering Barnes’ multiple Bernoulli polynomials as well as generalized Barnes’ multiple Frobenius-Euler polynomials, we define and investigate the mixed-type polynomials of these polynomials. From the properties of Sheffer sequences of these polynomials arising from umbral calculus, we derive new and interesting identities.

MSC: 05A15; 05A40; 11B68; 11B75; 33E20; 65Q05

1 Introduction

In this paper, we consider the polynomials

BHn(x) =BHn(x|a;b;λ;μ) =BHn(x|a, . . . ,ar;b, . . . ,bs;λ, . . . ,λs;μ, . . . ,μs)

called Barnes’ multiple Bernoulli and generalized Barnes’ multiple Frobenius-Euler mixed-type polynomials, whose generating function is given by

r

i=

t eait– 

s

j=

 –λj

ebjtλ

j μj

ext= ∞

n= BHn(x)

tn

n!, ()

wherea, . . . ,ar,b, . . . ,bs,λ, . . . ,λs,μ, . . . ,μs∈Cwitha, . . . ,ar,b, . . . ,bs= ,λ, . . . ,λs=

. Whenx= ,

BHn=BHn() =BHn(;a, . . . ,ar;b, . . . ,bs;λ, . . . ,λr;μ, . . . ,μr)

are called Barnes’ multiple Bernoulli and generalized Barnes’ multiple Frobenius-Euler mixed-type numbers.

Recall that Barnes’ multiple Bernoulli polynomials, denoted byBn(x|a, . . . ,ar), are given

by the generating function as

r

i=

t eait– 

ext= ∞

n=

Bn(x|a, . . . ,ar)

tn

n!, ()

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wherea, . . . ,ar=  [, ]. In addition, the generalized Barnes’ multiple Frobenius-Euler

polynomials, denoted byHn(x|b;λ;μ) =Hn(x|b, . . . ,bs;λ, . . . ,λs;μ, . . . ,μs), are given by

the generating function as

s

j=

 –λj

ebjtλ

j μj

ext= ∞

n=

Hn(x|b;λ;μ)

tn

n! ()

(seee.g.[–]). Ifμ=· · ·=μs= , then

Hn(x|b, . . . ,bs;λ, . . . ,λs) =Hn(x|b, . . . ,bs;λ, . . . ,λs; , . . . , )

are called Barnes-type Frobenius-Euler polynomials. If furtherλ=· · ·=λs=λandb=

· · ·=bs= , thenHn(s)(x|λ) =Hn(x|, . . . , ;λ, . . . ,λ; , . . . , ) are called Frobenius-Euler

poly-nomials of order s(seee.g.[, ]). Ifλ=· · ·=λs= –, thenEn(x|b, . . . ,bs;μ, . . . ,μs) =

Hn(x|b, . . . ,bs; , . . . , ;μ, . . . ,μs) are called generalized Barnes-type Euler polynomials.

These polynomials arise naturally in connection with the study of Barnes-type Peters

poly-nomials. Peters polynomials were mentioned in [, p.] and were investigated ine.g.

[].

In this paper, by considering Barnes’ multiple Bernoulli polynomials as well as general-ized Barnes’ multiple Frobenius-Euler polynomials, we define and investigate the mixed-type polynomials of these polynomials. From the properties of Sheffer sequences of these polynomials arising from umbral calculus, we derive new and interesting identities.

2 Umbral calculus

LetCbe a complex number field and letF be the set of all formal power series in the

variablet:

F=

f(t) = ∞

k= ak

k!t ka

k∈C

. ()

LetP=C[x] and letP∗be the vector space of all linear functionals onP.L|p(x)is the

action of the linear functionalLon the polynomialp(x), and we recall that the vector space

operations onP∗are defined byL+M|p(x)=L|p(x)+M|p(x),cL|p(x)=cL|p(x),

wherecis a complex constant inC. Forf(t)∈F, let us define the linear functional onP

by setting

f(t)|xn=a

n (n≥). ()

In particular,

tk|xn=n!δn,k (n,k≥), ()

whereδn,kis the Kronecker symbol.

ForfL(t) =

k=L|x k

k! tk, we havefL(t)|xn=L|xn. That is,L=fL(t). The mapLfL(t)

is a vector space isomorphism fromP∗ontoF. Henceforth,Fdenotes both the algebra

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elementf(t) ofFwill be thought of as both a formal power series and a linear functional.

We callFumbral algebra, andumbral calculusis the study of umbral algebra. The order

O(f(t)) of a power seriesf(t) (= ) is the smallest integerkfor which the coefficient oftk

does not vanish. IfO(f(t)) = , thenf(t) is called adelta series; ifO(f(t)) = , thenf(t) is

called aninvertible series. Forf(t),g(t)∈F withO(f(t)) =  andO(g(t)) = , there exists

a unique sequencesn(x) (degsn(x) =n) such thatg(t)f(t)k|sn(x)=n!δn,k forn,k≥ [,

Theorem ..]. Such a sequencesn(x) is called theSheffer sequencefor (g(t),f(t)) which is

denoted bysn(x)∼(g(t),f(t)).

Forf(t),g(t)∈Fandp(x)∈P, we have

f(t)g(t)|p(x)= f(t)|g(t)p(x)= g(t)|f(t)p(x), ()

and

f(t) = ∞

k=

f(t)|xkt

k

k!, p(x) = ∞

k=

tk|p(x)x k

k!, ()

[, Theorem ..]. Thus, by (), we get

tkp(x) =p(x) = d kp(x)

dxk and e

ytp(x) =p(x+y). ()

Sheffer sequences are characterized in the generating function [, Theorem ..].

Lemma  The sequence sn(x)is Sheffer for(g(t),f(t))if and only if

g(f¯(t))e y¯f(t)=

k= sk(y)

k! t

k (yC),

wheref¯(t)is the compositional inverse of f(t).

For sn(x)∼ (g(t),f(t)), we have the following equations [, Theorem ..,

Theo-rem .., TheoTheo-rem ..]:

f(t)sn(x) =nsn–(x) (n≥), ()

sn(x) = n

j=

j! g

¯

f(t)–f¯(t)j|xnxj, ()

sn(x+y) = n

j=

n j

sj(x)pnj(y), ()

wherepn(x) =g(t)sn(x).

Assume thatpn(x)∼(,f(t)) andqn(x)∼(,g(t)). Then the transfer formula [,

Corol-lary ..] is given by

qn(x) =x

f(t)

g(t)

n

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Forsn(x)∼(g(t),f(t)) andrn(x)∼(h(t),l(t)), assume that

sn(x) = n

m=

Cn,mrm(x) (n≥).

Then we have [, p.]

Cn,m= 

m!

hf(t))

g(f¯(t))l

¯

f(t)mxn

. ()

3 Main results

From definitions (), () and (),Bn(x|a, . . . ,ar),Hn(x|b, . . . ,bs;λ, . . . ,λs;μ, . . . ,μs) and

BHn(x|a, . . . ,ar;b, . . . ,bs;λ, . . . ,λs;μ, . . . ,μs) are the Appell sequences for

r

i=

eait–  t

, s

j=

ebjtλ

j

 –λj

μj

and r

i=

eait–  t

s

j=

ebjtλ

j

 –λj

μj

,

respectively. So,

Bn(x|a, . . . ,ar)∼

r

i=

eait–  t

,t

, ()

Hn(x|b, . . . ,bs;λ, . . . ,λs;μ, . . . ,μs)∼

s

j=

ebjtλ

j

 –λj

μj

,t

, ()

BHn(x|a, . . . ,ar;b, . . . ,bs;λ, . . . ,λs;μ, . . . ,μs) ∼

r

i=

eait–  t

s

j=

ebjtλ

j

 –λj

μj

,t

. ()

In particular,

tBn(x|a) =

d

dxBn(x|a) =nBn–(x|a),

tHn(x|b;λ;μ) =

d

dxHn(x|b;λ;μ) =nHn–(x|b;λ;μ), tBHn(x|a;b;λ;μ) =

d

dxBHn(x|a;b;λ;μ) =nBHn–(x|a;b;λ;μ),

wherea= (a, . . . ,ar),b= (b, . . . ,bs),λ= (λ, . . . ,λs) andμ= (μ, . . . ,μs).

3.1 Explicit expressions

Let (n)j=n(n– )· · ·(nj+ ) (j≥) with (n)= . The (signed) Stirling numbers of the

first kindS(n,m) are defined by

(x)n=

n

m=

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Notice that

(x)n

,et– . ()

Theorem  We have

BHn(x|a;b;λ;μ) =

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=

Thus, we obtain (). Finally, we obtain that

BHn(y|a;b;λ;μ) =

3.2 The Sheffer identity Theorem 

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3.3 Recurrence Theorem 

 – r

n+ 

BHn+(x|a;b;λ;μ)

=xBHn(x|a;b;λ;μ) –

n+  r

i=

aiBHn+(x+ai|a,ai;b;λ;μ)

s

j=

μjbj

 –λj

BHn(x+bj|a;b;λ;μ+ej). ()

Remark Whenn=r– , as the left-hand side of () is equal to , we have

xBHr–(x|a;b;λ;μ) =

r

r

i=

aiBHr(x+ai|a,ai;b;λ;μ)

+ s

j=

μjbj

 –λj

BHr–(x+bj|a;b;λ;μ+ej).

Proof By applying

sn+(x) =

xg (t)

g(t)

f(t)sn(x) ()

[, Corollary ..] with (), we get

BHn+(x|a;b;λ;μ) =xBHn(x|a;b;λ;μ) –

g(t)

g(t)BHn(x|a;b;λ;μ). Since

g(t)

g(t) =

lng(t)=

r

i=

lneait– rlnt+

s

j=

μjln

ebjtλ

j

s

j=

μjln( –λj)

= r

i= aieait

eait– r t+

s

j=

μjbjebjt

ebjtλ

j

=

t

r

i=

aiteait

eait– – 

+ s

j=

μjbjebjt

ebjtλ

j

and

r

i=

aiteait

eait– – 

=

r

i= ai

t+· · ·

is a series with order at least one, we have

g(t)

g(t)BHn(x|a;b;λ;μ) =

t

r

i=

aiteait

eait– – 

+ s

j=

μjbjebjt

 –λj

 –λj

ebjtλ

j

(8)

=

3.4 More relations

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=

The third term is

y

the first term is

(10)

Since

the second term is

Therefore, we obtain

BHn(x|a;b;λ;μ) =xBHn–(x|a;b;λ;μ) –

which is identity ().

3.5 A relation including Bernoulli numbers Theorem  For n– ≥m≥,we have

Proof We shall compute

r

in two different ways. On the one hand, it is equal to

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On the other hand, it is equal to

The second term of () is equal to

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Therefore, we get, forn– ≥m≥,

(n)mBHnm(a;b;λ;μ)

=m(n– )m–BHnm(a;b;λ;μ)

– (n– )m

s

j=

μjbj

 –λj

BHnm–(bj|a;b;λ;μ+ej)

+ (n– )m–

nm

l=

(–)l–

nm l

BlBHnml(a;b;λ;μ) r

i= ali.

Dividing both sides by (n– )m–, we obtain, forn– ≥m≥,

(nm)BHnm(a;b;λ;μ)

= –(nm)

s

j=

μjbj

 –λj

BHnm–(bj|a;b;λ;μ+ej)

+ nm

l=

(–)l–

nm l

BlBHnml(a;b;λ;μ) r

i= al

i.

Thus, we get ().

3.6 A relation with Stirling numbers

The Stirling numbers of the second kindS(n,m) are defined by

(et– )m

m! =

n=m

S(n,m) tn

n!.

Then

φn(x) := n

m=

S(n,m)xm∼,ln( +t). ()

Theorem 

BHn(x|a;b;λ;μ) = n

m=

n

l=m

n l

S(l,m)BHnlφm(x). ()

Proof For () and (), assume thatBHn(x|a;b;λ;μ) =

n

m=Cn,mφm(x). By (), we have

Cn,m= 

m!

r i=(e

ait–

t ) s

j=(

ebjtλj –λj )

μj

ln( +t)mxn

=

r

i=

t eait– 

s

j=

 –λj

ebjtλ

j μj

m!ln( +t)mxn

=

r

i=

t eait– 

s

j=

 –λj

ebjtλ

j μj

l=m

S(l,m)t l

l!x n

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=

3.7 A relation with falling factorials Theorem 

BHn(x|a;b;λ;μ) =

3.8 A relation with higher-order Frobenius-Euler polynomials Theorem 

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assume thatBHn(x|a;b;λ;μ) =

n

m=Cn,mHm(s)(x|α). By (), similarly to the proof of (),

we have

Cn,m=

3.9 A relation with higher-order Bernoulli polynomials

Bernoulli polynomialsB(np)(x) of orderpare defined by

Theorem 

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assume thatBHn(x|a;b;λ;μ) =

n

m=Cn,mB(mp)(x). By (), similarly to the proof of (), we

have

Cn,m= 

m!

(ett–)p r

i=(e ait–

t ) s

j=(

ebjtλj –λj )

μj tmxn

=

n m

r

i=

t eait– 

s

j=

 –λj

ebjtλ

j μj

ett– pxnm

=

n m

r

i=

t eait– 

s

j=

 –λj

ebjtλ

j μj

p! ∞

l=

S(l+p,p) t l

(l+p)!x

nm

=

n m

p! nm

l=

(nm)l

(l+p)! S(l+p,p)

r

i=

t eait– 

s

j=

 –λj

ebjtλ

j μj

xnml

=

n m

nm

l=

nm l

l+p

p

S(l+p,p)BHnml

=

n m

nm

l=

nm l

nml+p

p

S(nml+p,p)BHl.

Thus, we get identity ().

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

1Department of Mathematics, Sogang University, Seoul, 121-742, Republic of Korea.2Department of Mathematics,

Kwangwoon University, Seoul, 139-701, Republic of Korea.3Graduate School of Science and Technology, Hirosaki

University, Hirosaki, 036-8561, Japan.

Acknowledgements

The authors would like to thank the referees for their valuable comments. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MOE) (No. 2012R1A1A2003786) and was partially supported by Kwangwoon University in 2014.

Received: 9 June 2014 Accepted: 11 August 2014 Published:09 Sep 2014 References

1. Araci, S, Acikgoz, M: A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials. Adv. Stud. Contemp. Math.22, 399-406 (2012)

2. Ota, K: On Kummer-type congruences for derivatives of Barnes’ multiple Bernoulli polynomials. J. Number Theory92, 1-36 (2002)

3. Kim, DS, Kim, T: Higher-order Frobenius-Euler and poly-Bernoulli mixed-type polynomials. Adv. Differ. Equ.2013, 251 (2013)

4. Kim, DS, Kim, T, Lee, S-H, Rim, S-H: A note on the higher-order Frobenius-Euler polynomials and Sheffer sequences. Adv. Differ. Equ.2013, 41 (2013)

5. Can, M, Cenkci, M, Kurt, V, Simsek, Y: Twisted Dedekind type sums associated with Barnes’ type multiple Frobenius-Eulerl-functions. Adv. Stud. Contemp. Math.18, 135-160 (2009)

6. Ryoo, CS, Song, H, Agarwal, RP: On the roots of theq-analogue of Euler-Barnes’ polynomials. Adv. Stud. Contemp. Math.9, 153-163 (2004)

7. Simsek, Y, Yurekli, O, Kurt, V: On interpolation functions of the twisted generalized Frobenius-Euler numbers. Adv. Stud. Contemp. Math.15, 187-194 (2007)

8. Kim, DS, Kim, T: Some identities of Frobenius-Euler polynomials arising from umbral calculus. Adv. Differ. Equ.2012, 196 (2012)

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10. Roman, S: The Umbral Calculus. Dover, New York (2005)

11. Kim, DS, Kim, T: Poly-Cauchy and Peters mixed-type polynomials. Adv. Differ. Equ.2014, 4 (2014)

10.1186/1687-1847-2014-238

References

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