R E S E A R C H
Open Access
Barnes-type Narumi polynomials
Dae San Kim
1and Taekyun Kim
2**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 study the Barnes-type Narumi polynomials with umbral calculus viewpoint. From our study, we derive various identities of the Barnes-type Narumi polynomials.
MSC: 05A19; 05A40; 11B68
Keywords: Barnes-type Narumi polynomial; umbral calculus
1 Introduction
As is well known, the Narumi polynomials of orderαare defined by the generating
func-tion to be
t
log( +t)
α
( +t)x=
∞
n=
Nn(α)(x)t
n
n! (see []). ()
Letr∈Z>. We consider the polynomialsNn(x|a, . . . ,ar) andNˆn(x|a, . . . ,ar),
respec-tively, called the Barnes-type Narumi polynomials of the first kind and those of the second kind and respectively given by
r
j=
( +t)aj–
log( +t)
( +t)x=
∞
n=
Nn(x|a, . . . ,ar)
tn
n! ()
and
r
j=
( +t)aj–
log( +t)( +t)aj
( +t)x=
∞
n= ˆ
Nn(x|a, . . . ,ar)
tn
n!, ()
wherea,a, . . . ,ar= .
Whenx= ,
Nn(a, . . . ,ar) =Nn(|a, . . . ,ar)
and
ˆ
Nn(a, . . . ,ar) =Nˆn(|a, . . . ,ar)
are respectively called the Barnes-type Narumi numbers of the first kind and those of the second kind.
Note that
Nn(x|, . . . ,
r
) =Nn(r)(x),
ˆ
Nn(x|, . . . ,
r
) =Nˆn(r)(x)
and
ˆ
Nn(x|, . . . ,
r
) =Nn(r)(x–r).
In the previous paper [],Nn(α)(x) was denoted byNn(–α)and called the Narumi
polyno-mial of orderα.
The Bernoulli polynomials are defined by the generating function to be
t et– e
xt=
∞
n= Bn(x)
tn
n! (see [–]). ()
Whenx= ,Bn=Bn() are called the Bernoulli numbers. In [], it is known that the
Cauchy numbers are given by
t
log( +t)= ∞
n= Cn
tn
n!. ()
It is well known that the Stirling number of the first kind is given by
(x)n=x(x– )· · ·(x–n+ ) =
∞
l=
S(n,l)xl (n≥) (see [, , –]). ()
From (), we have
log( +t)n=n! ∞
l=n
S(l,n)t
l
l! (n≥). ()
LetCbe the complex number field and letFbe the set of all formal power series in the
variablet:
F=
f(t) =
∞
k= ak
tk
k!
ak∈C
.
LetP=C[x] and letP∗ be the vector space of all linear functionals onP.L|p(x)
de-notes the action of the linear functionalLonp(x) which satisfiesL+M|p(x)=L|p(x)+
M|p(x), andcL|p(x)=cL|p(x), wherecis a complex constant. The linear functional
f(t)|·onPis defined byf(t)|xn=a
n(n≥), wheref(t)∈F. Thus, we note that
tk|xn=n!δn,k (n,k≥), ()
ForfL(t) =
∞
k=
L|xk
k! tk, we havefL(t)|xn=L|xn. So, the mapL →fL(t) is a vector
space isomorphism fromP∗ ontoF. Henceforth,F denotes both the algebra of formal
power series intand the vector space of all linear functionals onP, and so an element
f(t) ofF will be thought of as both a formal power series and a linear functional. We
callFthe umbral algebra. The ordero(f(t)) of a power seriesf(t)= is the smallest
in-teger for which the coefficient of tk does not vanish. Ifo(f(t)) = , thenf(t) is called a
delta series; if o(f(t)) = , theng(t) is called an invertible series. Letf(t),g(t)∈F with
o(f(t)) = ando(g(t)) = . Then there exists a unique sequencesn(x) (degsn(x) =n) such
thatg(t)f(t)k|s
n(x)=n!δn,kforn,k≥. The sequencesn(x) is called the Sheffer sequence
for (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!.
()
From (), we can derive the following equation ():
tkp(x) =p(k)(x) =d
kp(x)
dxk , e
ytp(x) =p(x+y) (see []). ()
Letsn(x)∼(g(t),f(t)). Then the following will be used:
dsn(x)
dx =
n–
l=
n l
f(t)|xn–lsl(x), ()
wheref(t) is the compositional inverse off(t) withf(f(t)) =f(f(t)) =t,
g(f(t))e
xf(t)= ∞
n= sn(x)
tn
n! for allx∈C, ()
f(t)sn(x) =nsn–(x) (n≥), sn(x) = n
j=
g(f(t))–f(t)j|xn
j! x
j, ()
sn(x+y) = n
j=
n j
sj(x)pn–j(y), wherepn(x) =g(t)sn(x), ()
f(t)|xp(x)=∂tf(t)|p(x), where∂tf(t) = df(t)
dt ()
and
sn+(x) =
x–g (t)
g(t)
Let us assume thatsn(x)∼(g(t),f(t)) andrn(x)∼(h(t),l(t)). Then we have
sn(x) = n
m=
cn,mrm(x) (n≥), ()
where
cn,m=
m!
h(f(t)) g(f(t))l
f(t)mxn
(see [, ]). ()
From (), () and (), we note that
Nn(x|a, . . . ,ar)∼
r
j=
t eajt–
,et–
()
and
ˆ
Nn(x|a, . . . ,ar)∼
r
j=
teajt
eajt–
,et–
. ()
In this paper, we study the Barnes-type Narumi polynomials with umbral calculus view-point. From our study, we derive various identities of the Barnes-type Narumi polynomi-als.
2 Barnes-type Narumi polynomials
From (), we note that
r
j=
t eajt–
Nn(x|a, . . . ,ar)∼
,et– ()
and
(x)n∼
,et– . ()
Thus, by () and (), we get
Nn(x|a, . . . ,ar) = r
j=
eajt–
t
(x)n
=
n
m= S(n,m)
r
j=
eajt–
t
xm. ()
Note that
r
j=
eajt–
t
=
∞
l= al+ (l+ )!
tl
× · · · ×
∞
lr=
alr+
r
(lr+ )!
tlr
= ∞
l,...,lr=
l+···+lr=i
al+· · ·alr+
r
(l+ )!· · ·(lr+ )!
From () and (), we have Thus, we have
=
Therefore, by (), () and (), we obtain the following theorem.
Theorem For n≥,we have
From (), we can derive the following equation ():
Nn(x|a, . . . ,ar) =
Theorem For n≥,we have
Theorem For n≥,we have
Nn(x|a, . . . ,ar) = n
m=
n m
Nn–m(a, . . . ,ar)(x)m
and
ˆ
Nn(x|a, . . . ,ar) = n
m=
n m
ˆ
Nn–m(a, . . . ,ar)(x)m.
From (), we note that
Nn(x+y|a, . . . ,ar) = n
j=
n j
Nj(x|a, . . . ,ar)(y)n–j ()
and
ˆ
Nn(x+y|a, . . . ,ar) = n
j=
n j
ˆ
Nj(x|a, . . . ,ar)(y)n–j. ()
By (), we get
et– Nn(x|a, . . . ,ar) =nNn–(x|a, . . . ,ar) ()
and
et– Nn(x|a, . . . ,ar) =etNn(x|a, . . . ,ar) –Nn(x|a, . . . ,ar)
=Nn(x+ |a, . . . ,ar) –N(x|a, . . . ,ar). ()
From () and (), we have
Nn(x+ |a, . . . ,ar) –Nn(x|a, . . . ,ar) =nNn–(x|a, . . . ,ar). ()
By the same method as (), we get
ˆ
Nn(x+ |a, . . . ,ar) –Nˆn(x|a, . . . ,ar) =nNˆn–(x|a, . . . ,ar). ()
Recall thatNn(x|a, . . . ,ar)∼(
r
j=(eajtt–),et– ).
From (), we can derive the following equation ():
Nn+(x|a, . . . ,ar) =xNn(x– |a, . . . ,ar) –e–t
g(t)
g(t)Nn(x|a, . . . ,ar). ()
Now, we observe that
g(t) g(t) =
logg(t)=
rlogt–
r
j=
logeajt–
=r
t –
r
j= ajeajt
eajt–
=
r j=
r
i=j(eait– ){eajt– –tajeajt}
where
has at least the order . By () and (), we get
Therefore, by () and (), we obtain the following theorem.
By the same method as the proof of Theorem , we get
ˆ
Nn+(x|a, . . . ,ar) =
x–
r
j= aj
ˆ
Nn(x– |a, . . . ,ar)
+
n
m=
n
i=m n
l=i r
j= i+
n l
i+
m
S(l,i)
×Bi+–m(–aj)i+–mNˆn–l(a, . . . ,ar)
(x– )m. ()
From () and (), we can derive the following equation ():
f(t)|xn–l=log( +t)|xn–l=
∞
m=
(–)m–
m t
mxn–l
= (–)n–l–(n–l– )!. ()
Thus, by (), we get
d
dxNn(x|a, . . . ,ar) =
n–
l=
n l
(–)n–l–(n–l– )!Nl(x|a, . . . ,ar)
=n!
n–
l=
(–)n–l–
l!(n–l)Nl(x|a, . . . ,ar). ()
By the same method as (), we get
d
dxNˆn(x|a, . . . ,ar) =n!
n–
l=
(–)n–l–
l!(n–l)Nˆl(x|a, . . . ,ar). ()
From (), we note that, forn≥,
Nn(y|a, . . . ,ar)
=
∞
i=
Ni(y|a, . . . ,ar)
ti
i!
xn
=
r
j=
( +t)aj–
log( +t)
( +t)yxn
=
∂t
r
j=
( +t)aj–
log( +t)
( +t)y
xn–
=
r
j=
( +t)aj–
log( +t)
∂t( +t)yxn–
+
∂t
( +t)aj–
log( +t)
( +t)yxn–
=yNn–(y– |a, . . . ,ar) +
∂t
( +t)aj–
log( +t)
( +t)yxn–
Now, we observe that
Therefore, by () and (), we obtain the following theorem.
Theorem For n≥,we have
Nn(x|a, . . . ,ar) =xNn–(x– |a, . . . ,ar)
+
n
r
j=
n
l=
n l
ajClNn–l(x+aj– |a, . . . ,aˆj, . . . ,ar)
– r
n
r
l=
n l
ClNn–l(x– |a, . . . ,ar),
where Cnare the Cauchy numbers with the generating function given by
t
log( +t)= ∞
n= Cn
tn
n!.
By the same method as the proof of Theorem , we get
ˆ
Nn(x|a, . . . ,ar) =
x–
r
j= aj
ˆ
Nn–(x– |a, . . . ,ar)
– r
n
n
l=
n l
ClNˆn–l(x– |a, . . . ,ar)
–
n
r
j=
n
l=
n l
ajClNˆn–l(x– |a, . . . ,aˆj, . . . ,ar). ()
Now we compute the following formula () in two different ways:
r
j=
( +t)aj–
log( +t)
log( +t)mxn
. ()
On the one hand,
r
j=
( +t)aj–
log( +t)
log( +t)mxn
=
r
j=
( +t)aj–
log( +t)
m!
∞
l=m
S(l,m)t
l
l!x
n
=m!
n
l=m
n l
S(l,m)
r
j=
( +t)aj–
log( +t)
xn–l
=m!
n
l=m
n l
S(l,m)Nn–l(a, . . . ,ar)
=m!
n–m
l=
n l
On the other hand,
Note that
×
Therefore, by (), (), () and (), we obtain the following theorem.
Theorem For n– ≥m≥,we have
By the same method as the proof of Theorem , we get
Let us consider the following two Sheffer sequences:
Nn(x|a, . . . ,ar)∼
r
j=
t eajt–
,et–
()
and (). We let
Nn(x|a, . . . ,ar) = n
m=
Cn,m(x)m. ()
From () and (), we note that
Cn,m=
m!
r
j=
( +t)aj–
log( +t)
tmxn
=
n m
r
j=
( +t)aj–
log( +t)
xn–m
=
n m
Nn–m(a, . . . ,ar). ()
Therefore, by () and (), we obtain the following theorem.
Theorem For n≥,we have
Nn(x|a, . . . ,ar) = n
m=
n m
Nn–m(a, . . . ,ar)(x)m.
By the same method as the proof of Theorem , we get
ˆ
Nn(x|a, . . . ,ar) = n
m=
n m
ˆ
Nn–m(a, . . . ,ar)(x)m. ()
For
Nn(x|a, . . . ,ar)∼
r
j=
t eajt–
,et–
and
Hn(s)(x|λ)∼
et–λ
–λ
s
,t
, λ∈Cwithλ= ,
let us assume that
Nn(x|a, . . . ,ar) = n
m=
whereHm(s)(x|λ) are the Frobenius-Euler polynomials of ordersdefined by the generating
function as
–λ
et–λ
s
ext= ∞
n=
Hn(s)(x|λ)t
n
n!.
From () and (), we note that
Cn,m=
m!( –λ)s
r
j=
( +t)aj–
log( +t)
log( +t)m( –λ+t)sxn
=
m!( –λ)s
r
j=
( +t)aj–
log( +t)
log( +t)m
min{s,n}
j=
s j
( –λ)s–jtjxn
=
m!( –λ)s
n–m
j=
s j
( –λ)s–j(n)j
× r
j=
( +t)aj–
log(t)
log( +t)mxn–j
=
m!( –λ)s
n–m
j=
s j
( –λ)s–j(n)jm!
×
n–j–m
l=
n–j l
S(n–j–l,m)Nl(a, . . . ,ar)
=
n–m
j=
n–m–j
l=
s j
n–j
l
×(n)j( –λ)–jS(n–j–l,m)Nl(a, . . . ,ar). ()
Therefore, by () and (), we obtain the following theorem.
Theorem For n≥,we have
Nn(x|a, . . . ,ar)
=
n
m= n–m
j=
n–m–j
l=
s j
n–j
l
(n)j( –λ)–j
×S(n–j–l,m)Nl(a, . . . ,ar)
Hm(s)(x|λ).
By the same method as the proof of Theorem , we get
ˆ
Nn(x|a, . . . ,ar)
=
n
m= n–m
j=
n–m–j
l=
s j
n–j
l
(n)j
×( –λ)–jS(n–j–l,m)Nˆl(a, . . . ,ar)
Now, we consider the following two Sheffer sequences:
whereB(ns)(x) are the Bernoulli polynomials of ordersgiven by the generating function as
Let us assume that
Nn(x|a, . . . ,ar) =
whereC(ks) are the Cauchy numbers of the first kind of ordersdefined by the generating
function as
Theorem For n≥,we have
Nn(x|a, . . . ,ar) = n
m= n–m
k=
n–m–k
l=
n k
n–k
l
C(s)
k
×S(n–k–l,m)Nl(a, . . . ,ar)
B(ms)(x).
By the same method as the proof of Theorem , we get
ˆ
Nn(x|a, . . . ,ar) = n
m= n–m
k=
n–m–k
l=
n k
n–k
l
C(s)
k
×S(n–k–l,m)Nˆl(a, . . . ,ar)
B(ms)(x).
Recently, several authors have studied umbral calculus (see [–, –, ]).
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, 121-742, Republic of Korea.2Department of Mathematics,
Kwangwoon University, Seoul, 139-701, Republic of Korea.
Acknowledgements
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MOE) (No. 2012R1A1A2003786). The authors would like to thank the referees for their valuable comments.
Received: 30 April 2014 Accepted: 9 June 2014 Published:22 Jul 2014
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