R E S E A R C H
Open Access
Barnes’ multiple Frobenius-Euler and
poly-Bernoulli mixed-type polynomials
Dae San Kim
1, Taekyun Kim
2*, Jong-Jin Seo
3and Takao Komatsu
4*Correspondence: [email protected] 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, we consider Barnes’ multiple Frobenius-Euler and poly-Bernoulli mixed-type 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; 65Q05
1 Introduction
In this paper, we consider the polynomialsTn(r,k)(x|a, . . . ,ar;λ, . . . ,λr) whose generating
function is given by
r
j=
–λj
eajt–λ
j
Lik( –e–t)
–e–t e
xt=
∞
n=
Tn(r,k)(x|a, . . . ,ar;λ, . . . ,λr)
tn
n!, ()
wherer∈Z>,k∈Z,a, . . . ,ar= ,λ, . . . ,λr= and
Lik(x) = ∞
m=
xm mk
is the kth polylogarithm function. Tn(r,k)(x|a, . . . ,ar;λ, . . . ,λr) will be called Barnes’
multiple Frobenius-Euler and poly-Bernoulli mixed-type polynomials. When x = , Tn(r,k)(a, . . . ,ar;λ, . . . ,λr) = Tn(r,k)(|a, . . . ,ar;λ, . . . ,λr) will be called Barnes’ multiple
Frobenius-Euler and poly-Bernoulli mixed-type numbers.
Recall that, for every integerk, the poly-Bernoulli polynomialsB(nk)(x) are defined by the
generating function as follows:
Lik( –e–t)
–e–t e
xt=
∞
n=
B(nk)(x)t
n
n! ()
([],cf.[]). Also, as a natural generalization of higher-order Frobenius-Euler polynomials, Barnes’ multiple Frobenius-Euler polynomialsHn(r)(x|a, . . . ,ar;λ, . . . ,λr) are defined by
the generating function as follows:
r
j=
–λj
eajt–λ
j
ext=
∞
n=
Hn(r)(x|a, . . . ,ar;λ, . . . ,λr)
tn
n!, ()
wherea, . . . ,ar= . Note that the Frobenius-Euler polynomials of orderr,Hn(r)(x|λ) are
defined by the generating function
–λ
et–λ
r
ext=
∞
n=
Hn(r)(x|λ)t
n
n!
(see,e.g., []).
In this paper, we consider Barnes’ multiple Frobenius-Euler and poly-Bernoulli mixed-type polynomials. From the properties of Sheffer sequences of these polynomials arising from umbral calculus, we derive new and interesting identities.
2 Umbral calculus
LetCbe the 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=an (n≥). ()
In particular,
tk|xn=n!δn,k (n,k≥), ()
whereδn,kis the Kronecker symbol.
ForfL(t) =
∞ k=L|x
k
k! t
k, we havef
L(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 of formal power series intand the vector space of all linear functionals onP, and so an elementf(t) ofFwill be thought of as both a formal power series and a linear functional. We callFtheumbral algebraand theumbral calculusis the study of umbral algebra. The orderO(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)) = , then
f(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,kforn,k≥.
Such a sequence sn(x) is called the Sheffer sequencefor (g(t),f(t)) which is denoted by
sn(x)∼(g(t),f(t)).
Forf(t),g(t)∈Fandp(x)∈P, we have
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(k)(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
x–pn(x) (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
We now note that B(nk)(x), Hn(r)(x|a, . . . ,ar;λ, . . . ,λr) andTn(r,k)(x|a, . . . ,ar;λ, . . . ,λr) are
the Appell sequences for
gk(t) =
–e–t
Lik( –e–t)
, gr(t) = r
j=
eajt–λ
j
–λj
,
gr,k(t) = r
j=
eajt–λ
j
–λj
–e–t
Lik( –e–t)
.
So,
B(nk)(x)∼
–e–t Lik( –e–t),t
, ()
Hn(r)(x|a, . . . ,ar;λ, . . . ,λr)∼
r
j=
eajt–λ
j
–λj
,t
, ()
Tn(r,k)(x|a, . . . ,ar;λ, . . . ,λr)∼
r
j=
eajt–λ
j
–λj
–e–t
Lik( –e–t)
,t
. ()
In particular, we have
tB(nk)(x) = d dxB
(k)
n (x) =nB
(k)
n–(x), ()
tHn(r)(x|a, . . . ,ar;λ, . . . ,λr) =
d dxH
(r)
n (x|a, . . . ,ar;λ, . . . ,λr)
=nHn(r–)(x|a, . . . ,ar;λ, . . . ,λr), ()
tTn(r,k)(x|a, . . . ,ar;λ, . . . ,λr) =
d dxT
(r,k)
n (x|a, . . . ,ar;λ, . . . ,λr)
=nTn(r–,k)(x|a, . . . ,ar;λ, . . . ,λr). ()
Notice that
d
dxLik(x) =
xLik–(x).
3.1 Explicit expressions
WriteHn(r)(a, . . . ,ar;λ, . . . ,λr) :=Hn(r)(|a, . . . ,ar;λ, . . . ,λr). Let (n)j=n(n– )· · ·(n–j+ )
(j≥) with (n)= .
Theorem
Tn(r,k)(x|a, . . . ,ar;λ, . . . ,λr)
=
n
l=
n l
B(lk)(x)Hn(r–)l(a, . . . ,ar;λ, . . . ,λr) ()
=
n
l=
n l
= We also have
= In [] we obtained that
Lik( –e–t)
which is identity (). In [] we obtained that
Lik( –e–t)
whereS(l,m) are the Stirling numbers of the second kind, defined by
=
n
j=
n–j
m=
(–)n–m–j (m+ )k
n j
m!S(n–j,m)
Hj(r)(x|a, . . . ,ar;λ, . . . ,λr)
=
n
j=
n–j
m=
(–)n–m–j (m+ )k
n j
m!S(n–j,m)
j
l=
j l
Hj(–r)l(a, . . . ,ar;λ, . . . ,λr)xl
=
n
l=
n
j=l n–j
m=
(–)n–m–j
n j
j l
m!
(m+ )kS(n–j,m)H
(r)
j–l(a, . . . ,ar;λ, . . . ,λr)
xl,
which is identity (). By () with (), we have
gf¯(t)–f¯(t)j|xn=
r
j=
–λj
eajt–λ
j
Lik( –e–t)
–e–t t jxn
= (n)j
r
j=
–λj
eajt–λ
j
Lik( –e–t)
–e–t
xn–j
= (n)j
∞
i=
Ti(r,k)(a, . . . ,ar;λ, . . . ,λr)
ti i!
xn–j
= (n)jTn(r–,kj)(a, . . . ,ar;λ, . . . ,λr).
Thus, we get ().
3.2 Sheffer identity Theorem
Tn(r,k)(x+y|a, . . . ,ar;λ, . . . ,λr) = n
j=
n j
Tj(r,k)(x|a, . . . ,ar;λ, . . . ,λr)yn–j. ()
Proof By () with
pn(x) = r
j=
eajt–λ
j
–λj
–e–t
Lik( –e–t)
Tn(r,k)(x|a, . . . ,ar;λ, . . . ,λr)
=xn∼(,t),
using (), we have ().
3.3 Recurrence Theorem
Tn(r+,k)(x|a, . . . ,ar;λ, . . . ,λr) =xTn(r,k)(x|a, . . . ,ar;λ, . . . ,λr)
–
r
j= aj
–λj
Tn(r+,k)(x+aj|a, . . . ,ar,aj;λ, . . . ,λr,λj)
– n+
n+
l=
n+ l
×Tl(r,k)(x|a, . . . ,ar;λ, . . . ,λr)
–Tl(r,k–)(x|a, . . . ,ar;λ, . . . ,λr)
, ()
where Bnis the nth ordinary Bernoulli number.
Proof By applying
sn+(x) =
x–g
(t)
g(t)
f(t)sn(x)
[, Corollary ..] with (), we get
Tn(r+,k)(x|a, . . . ,ar;λ, . . . ,λr) =
x–g
r,k(t)
gr,k(t)
Tn(r,k)(x|a, . . . ,ar;λ, . . . ,λr).
Now,
gr,k(t) gr,k(t)
=lngr,k(t)
=
r
j=
lneajt–λ
j
–
r
j=
ln( –λj) +ln
–e–t–ln Lik
–e–t
=
r
j= ajeajt
eajt–λ
j
+ e –t
–e–t
–Lik–( –e –t)
Lik( –e–t)
=
r
j= ajeajt
eajt–λ
j
+ t et–
Lik( –e–t) –Lik–( –e–t)
tLik( –e–t)
.
Since
Tn(r,k)(x|a, . . . ,ar;λ, . . . ,λr) = r
i=
–λi
eait–λi
Lik( –e–t)
–e–t x n,
we have
Tn(r+,k)(x|a, . . . ,ar;λ, . . . ,λr)
=xTn(r,k)(x|a, . . . ,ar;λ, . . . ,λr)
–
r
j= ajeajt
–λj
–λj
eajt–λ
j r
i=
–λi
eait–λ
i
Lik( –e–t)
–e–t x n
– t et–
r
i=
–λi
eait–λi
Lik( –e–t) –Lik–( –e–t) t( –e–t) x
n.
Since
Lik( –e–t) –Lik–( –e–t)
–e–t =
k –
k–
is a delta series, we get
Lik( –e–t) –Lik–( –e–t)
t( –e–t) x
n=
n+
Lik( –e–t) –Lik–( –e–t)
–e–t x n+.
Therefore, by
t et– x
n+=B
n+(x) = n+
l=
n+ l
Bn+–lxl,
we obtain
Tn(r+,k)(x|a, . . . ,ar;λ, . . . ,λr)
=xTn(r,k)(x|a, . . . ,ar;λ, . . . ,λr) – r
j= aj
–λj
Tn(r+,k)(x+aj|a, . . . ,ar,aj;λ, . . . ,λr,λj)
– n+
r
i=
–λi
eait–λi
Lik( –e–t) –Lik–( –e–t)
–e–t
t et– x
n+
=xTn(r,k)(x|a, . . . ,ar;λ, . . . ,λr) – r
j= aj
–λj
Tn(r+,k)(x+aj|a, . . . ,ar,aj;λ, . . . ,λr,λj)
– n+
n+
l=
n+ l
Bn+–l
×Tl(r,k)(x|a, . . . ,ar;λ, . . . ,λr) –Tl(r,k–)(x|a, . . . ,ar;λ, . . . ,λr)
,
which is identity ().
3.4 A more relation Theorem
Tn(r,k)(x|a, . . . ,ar;λ, . . . ,λr)
=xTn(r–,k)(x|a, . . . ,ar;λ, . . . ,λr)
–
r
j= aj
–λj
Tn(r–+,k)(x+aj|a, . . . ,ar,aj;λ, . . . ,λr,λj)
– n
n
l=
n l
Bn–l
Tl(r,k)(x|a, . . . ,ar;λ, . . . ,λr)
–Tl(r,k–)(x|a, . . . ,ar;λ, . . . ,λr)
. ()
Proof Forn≥, we have
Tn(r,k)(y|a, . . . ,ar;λ, . . . ,λr) =
∞
l=
Tl(r,k)(y|a, . . . ,ar;λ, . . . ,λr)
tl
l!
xn
=
r
j=
–λj
eajt–λ
j
Lik( –e–t)
–e–t e ytxn
Since
and the fact that
Lik–( –e–t) –Lik( –e–t)
is a delta series, we have
r
Therefore, we obtain the desired result.
Remark Afternis replaced byn+ , identity () becomes the recurrence formula ().
3.5 Relations with poly-Bernoulli numbers and Barnes’ multiple Bernoulli numbers
Proof We shall compute
in two different ways. On the one hand,
On the other hand,
3.6 Relations with the Stirling numbers of the second kind and the falling
=
3.8 Relations with higher-order Frobenius-Euler polynomials Theorem
3.9 Relations with higher-order Bernoulli polynomials Bernoulli polynomialsB(nr)(x) of orderrare defined by
Theorem
Tn(r,k)(x|a, . . . ,ar;λ, . . . ,λr)
=
n
m=
n m
n–m
l=
n–m l
l+s
l
S(l+s,s)Tn(r–,mk)–l(a, . . . ,ar;λ, . . . ,λr)
B(ms)(x). ()
Proof For () and
B(ns)(x)∼
et–
t
s
,t
,
assume thatTn(r,k)(x|a, . . . ,ar;λ, . . . ,λr) =
n
m=Cn,mB(ms)(x). By (), we have
Cn,m=
m!
et–
t
sr
j=
–λj
eajt–λ
j
Lik( –e–t)
–e–t t
mxn
=
n m
r
j=
–λj
eajt–λ
j
Lik( –e–t)
–e–t
ett– sxn–m
=
n m
r
j=
–λj
eajt–λ
j
Lik( –e–t)
–e–t
n–m
l= s!
(l+s)!S(l+s,s)t
lxn–m
=
n m
n–m
l= s!
(l+s)!S(l+s,s)(n–m)l
r
j=
–λj
eajt–λ
j
Lik( –e–t)
–e–t
xn–m–l
=
n m
n–m
l= s!
(l+s)!S(l+s,s)(n–m)lT (r,k)
n–m–l(a, . . . ,ar;λ, . . . ,λr)
=
n m
n–m
l=
n–m
l
l+s
l
S(l+s,s)Tn(r–,km)–l(a, . . . ,ar;λ, . . . ,λr).
Thus, we get identity ().
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the manuscript and typed, read, and approved the final manuscript.
Author details
1Department of Mathematics, Sogang University, Seoul, 121-741, Republic of Korea.2Department of Mathematics,
Kwangwoon University, Seoul, 139-701, Republic of Korea.3Department of Applied Mathematics, Pukyong National
University, Pusan, 608-739, Republic of Korea.4Graduate School of Science and Technology, Hirosaki University, Hirosaki,
036-8561, Japan.
Acknowledgements
The authors would like to thank the referees for their valuable comments and suggestions. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MOE) (No.
2012R1A1A2003786). The fourth author was supported in part by the Grant-in-Aid for Scientific research (C) (No. 22540005), the Japan Society for the Promotion of Science.
References
1. Bayad, A, Hamahata, Y: Polylogarithms and poly-Bernoulli polynomials. Kyushu J. Math.65, 15-24 (2011) 2. Coppo, M-A, Candelpergher, B: The Arakawa-Kaneko zeta functions. Ramanujan J.22, 153-162 (2010)
3. Kim, DS, Kim, T: Some identities of Frobenius-Euler polynomials arising from umbral calculus. Adv. Differ. Equ.2012, Article ID 196 (2012)
4. Roman, S: The Umbral Calculus. Dover, New York (2005)
5. Kim, DS, Kim, T, Lee, S-H: Poly-Bernoulli polynomials arising from umbral calculus. Available at http://arxiv.org/pdf/1306.6697.pdf
10.1186/1687-1847-2014-92
