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
3and 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
ebjt–λ
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!, ()
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
ebjt–λ
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 mapL →fL(t)
is a vector space isomorphism fromP∗ontoF. Henceforth,Fdenotes both the algebra
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 (y∈C),
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)pn–j(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
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!
h(¯f(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=
ebjt–λ
j
–λj
μj
and r
i=
eait– t
s
j=
ebjt–λ
j
–λj
μj
,
respectively. So,
Bn(x|a, . . . ,ar)∼
r
i=
eait– t
,t
, ()
Hn(x|b, . . . ,bs;λ, . . . ,λs;μ, . . . ,μs)∼
s
j=
ebjt–λ
j
–λj
μj
,t
, ()
BHn(x|a, . . . ,ar;b, . . . ,bs;λ, . . . ,λs;μ, . . . ,μs) ∼
r
i=
eait– t
s
j=
ebjt–λ
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– )· · ·(n–j+ ) (j≥) with (n)= . The (signed) Stirling numbers of the
first kindS(n,m) are defined by
(x)n=
n
m=
Notice that
(x)n∼
,et– . ()
Theorem We have
BHn(x|a;b;λ;μ) =
=
Thus, we obtain (). Finally, we obtain that
BHn(y|a;b;λ;μ) =
3.2 The Sheffer identity Theorem
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) =
x–g (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
ebjt–λ
j
– s
j=
μjln( –λj)
= r
i= aieait
eait– – r t+
s
j=
μjbjebjt
ebjt–λ
j
=
t
r
i=
aiteait
eait– –
+ s
j=
μjbjebjt
ebjt–λ
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
ebjt–λ
j
=
3.4 More relations
=
The third term is
y
the first term is
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
On the other hand, it is equal to
The second term of () is equal to
Therefore, we get, forn– ≥m≥,
(n)mBHn–m(a;b;λ;μ)
=m(n– )m–BHn–m(a;b;λ;μ)
– (n– )m
s
j=
μjbj
–λj
BHn–m–(bj|a;b;λ;μ+ej)
+ (n– )m–
n–m
l=
(–)l–
n–m l
BlBHn–m–l(a;b;λ;μ) r
i= ali.
Dividing both sides by (n– )m–, we obtain, forn– ≥m≥,
(n–m)BHn–m(a;b;λ;μ)
= –(n–m)
s
j=
μjbj
–λj
BHn–m–(bj|a;b;λ;μ+ej)
+ n–m
l=
(–)l–
n–m l
BlBHn–m–l(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)BHn–lφm(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=(
ebjt–λj –λj )
μj
ln( +t)mxn
=
r
i=
t eait–
s
j=
–λj
ebjt–λ
j μj
m!ln( +t)mxn
=
r
i=
t eait–
s
j=
–λj
ebjt–λ
j μj
∞
l=m
S(l,m)t l
l!x n
=
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
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
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=(
ebjt–λj –λj )
μj tmxn
=
n m
r
i=
t eait–
s
j=
–λj
ebjt–λ
j μj
ett– pxn–m
=
n m
r
i=
t eait–
s
j=
–λj
ebjt–λ
j μj
p! ∞
l=
S(l+p,p) t l
(l+p)!x
n–m
=
n m
p! n–m
l=
(n–m)l
(l+p)! S(l+p,p)
r
i=
t eait–
s
j=
–λj
ebjt–λ
j μj
xn–m–l
=
n m
n–m
l=
n–m l
l+p
p
S(l+p,p)BHn–m–l
=
n m
n–m
l=
n–m l
n–m–l+p
p
S(n–m–l+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)
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