R E S E A R C H
Open Access
Hermite and poly-Bernoulli mixed-type
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 consider Hermite and poly-Bernoulli mixed-type polynomials and investigate the properties of those polynomials which are derived from umbral calculus. Finally, we give various identities associated with Stirling numbers, Bernoulli and Frobenius-Euler polynomials of higher order.
1 Introduction
Forr∈Z≥, as is well known, the Bernoulli polynomials of orderrare defined by the generating function to be
∞
n=
B(r) n(x)
n! t n=
t et–
r
ext (see [–]). (.)
Fork∈Z, the polylogarithm is defined by
Lik(x) =
∞
n=
xn
nk. (.)
Note thatLi(x) = –log( –x).
The poly-Bernoulli polynomials are defined by the generating function to be
Lik( –e–t) –e–t e
xt=
∞
n=
B(nk)(x)t n
n! (see [, ]). (.)
Whenx= ,B(nk)=B(nk)() are called the poly-Bernoulli numbers (of indexk).
Forν(= )∈R, the Hermite polynomials of orderνare given by the generating function to be
e–νt
ext=
∞
n=
Hn(ν)(x)t n
n! (see [, , ]). (.)
Whenx= ,Hn(ν)=Hn(ν)() are called the Hermite numbers of orderν.
In this paper, we consider the Hermite and poly-Bernoulli mixed-type polynomials
HB(nν,k)(x) which are defined by the generating function to be
e–νt
Lik( –e –t)
–e–t e xt=
∞
n=
HB(nν,k)(x)t n
n!, (.)
wherek∈Zandν(= )∈R.
Whenx= ,HBn(ν,k)=HB(nν,k)() are called the Hermite and poly-Bernoulli mixed-type numbers.
LetFbe the set of all formal power series in the variabletoverCas follows:
F=
f(t) =
∞
k=
ak
tk k!
ak∈C
. (.)
LetP=C[x] andP∗denote the vector space of all linear functionals onP.
L|p(x)denotes the action of the linear functionalLon the polynomialp(x), and we re-call 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 lin-ear functional onPby setting
f(t)|xn =an (n≥). (.)
Then, by (.) and (.), we get
tk|xn =n!δn,k (n,k≥), (.)
whereδn,kis the Kronecker symbol.
ForfL(t) =∞k=Lk|x!ktk, 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 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 callFthe umbral algebra and the umbral calculus is the study of umbral algebra. The orderO(f) of the power seriesf(t)= is the smallest integer for whichakdoes not vanish. IfO(f) = , thenf(t) is called an invertible series. IfO(f) = , thenf(t) is called a delta series. Forf(t),g(t)∈F, we have
f(t)g(t)|p(x) =f(t)|g(t)p(x) =g(t)|f(t)p(x). (.)
Letf(t)∈Fandp(x)∈P. Then we have
f(t) =
∞
k=
f(t)|xk k! t
k, p(x) =
∞
k=
tk|p(x)
k! x
k (see [, , , , ]). (.)
By (.), we get
p(k)() =tk|p(x) =|p(k)(x), (.)
From (.), we have
tkp(x) =p(k)(x) =d kp(x)
dxk (see [, , ]). (.)
By (.), we easily get
eytp(x) =p(x+y), eyt|p(x) =p(y). (.)
ForO(f(t)) = ,O(g(t)) = , there exists a unique sequencesn(x) of polynomials such that g(t)f(t)k|xn=n!δn,k(n,k≥).
The sequencesn(x) is called the Sheffer sequence for (g(t),f(t)) which is denoted by
sn(x)∼(g(t),f(t)).
Letp(x)∈P,f(t)∈F. Then we see that
f(t)|xp(x) =∂tf(t)|p(x) =
df(t)
dt p(x)
. (.)
Forsn(x)∼(g(t),f(t)), we have the following equations:
h(t) =
∞
k=
h(t)|sk(x)
k! g(t)f(t)
k, p(x) =
∞
k=
g(t)f(t)k|p(x)
k! sk(x), (.)
whereh(t)∈F,p(x)∈P,
g(f¯(t))e y¯f(t)=
∞
n=
sn(y)t n
n!, (.)
wheref¯(t) is the compositional inverse forf(t) withf(f¯(t)) =t,
sn(x+y) = n
k=
n k
sk(y)pn–k(x), wherepn(x) =g(t)sn(x), (.)
f(t)sn(x) =nsn–(x), sn+(x) =
x–g
(t)
g(t)
f(t)sn(x), (.)
and the conjugate representation is given by
sn(x) = n
j=
j!
gf¯(t)–f¯(t)j|xnxj. (.)
Forsn(x)∼(g(t),f(t)),rn(x)∼(h(t),l(t)), we have
sn(x) = n
m=
Cn,mrm(x), (.)
where
Cn,m=
m!
h(¯f(t))
g(f¯(t))l
¯ f(t)mxn
In this paper, we consider Hermite and poly-Bernoulli mixed-type polynomials and in-vestigate the properties of those polynomials which are derived from umbral calculus. Finally, we give various identities associated with Bernoulli and Frobenius-Euler polyno-mials of higher order.
2 Hermite and poly-Bernoulli mixed-type polynomials
From (.) and (.), we note that
HB(nν,k)(x)∼
eνt
–e
–t
Lik( –e–t),t
, (.)
and, by (.), (.) and (.), we get
B(nk)(x)∼
–e–t
Lik( –e–t),t
, (.)
Hn(ν)(x)∼eνt
,t, wheren≥. (.)
From (.), (.), (.) and (.), we have
tB(nk)(x) =nB(nk–) (x), tHn(ν)(x) =nHn(ν–)(x), tHB(nν,k)(x) =nHB(nν–,k)(x). (.)
By (.), (.) and (.), we get
HB(nν,k)(x) =e–νt
Lik( –e –t)
–e–t x
n=e–νtB(k) n (x)
= [n]
m=
m!
–ν
m
(n)mB(nk–) m(x)
= [n]
m=
n
m
(m)!
m!
–ν
m
B(nk–)m(x). (.)
Therefore, by (.), we obtain the following proposition.
Proposition For n≥,we have
HB(nν,k)(x) = [n]
m=
n
m
(m)!
m!
–ν
m
B(nk–)m(x).
From (.), we can also derive
HB(nν,k)(x) = Lik( –e –t)
–e–t e
–νtxn=Lik( –e –t)
–e–t H (ν) n (x) =
∞
m=
( –e–t)m (m+ )k H
(ν) n (x)
= n
m= (m+ )k
m
j=
m
j
(–)je–jtHn(ν)(x)
= n
m= (m+ )k
m
j=
m
j
Therefore, by (.), we obtain the following theorem.
Therefore, by (.), we obtain the following theorem.
Theorem For n≥,we have Therefore, by (.), we obtain the following theorem.
From (.) and (.), we have
Therefore, by (.), we obtain the following theorem.
Theorem For n≥,we have
We note that
HB(nν,k)(x)∼
Now, we observe that
By (.) and (.), we get
HB(nν+,k)(x)
=xHB(nν,k)(x) –g
(t)
g(t)HB (ν,k) n (x)
=xHB(nν,k)(x) –νnHBn(ν–,k)(x) –e–νt
t
et–
Lik( –e–t) –Lik–( –e–t)
t( –e–t) x
n. (.)
It is easy to show that
Lik( –e–t) –Li
k–( –e–t)
–e–t =
∞
m=
mk –
mk–
–e–tm–
=
k –
k–
t+· · ·. (.)
Thus, by (.), we get
Lik( –e–t) –Li
k–( –e–t)
t( –e–t) x
n=Lik( –e–t) –Lik–( –e–t) –e–t
xn+
n+ . (.)
From (.), we can derive
e–νt
t
et–
Lik( –e–t) –Li
k–( –e–t)
t( –e–t) x n
=
n+
∞
l=
Bl
l!t l
HB(nν+,k)(x) –HB(nν+,k–)(x)
=
n+ n+
l=
Bl
l!t lHB(ν,k)
n+(x) –HB (ν,k–) n+ (x)
=
n+ n+
l=
n+
l
Bl
HBn(ν+–,k)l(x) –HBn(ν+–,k–)l (x). (.)
Therefore, by (.) and (.), we obtain the following theorem.
Theorem For n≥,we have
HB(nν+,k)(x)
=xHB(nν,k)(x) –νnHB(nν–,k)(x)
–
n+ n+
l=
n+
l
Bl
HBn(ν+–,k)l(x) –HBn(ν+–,k–)l (x). (.)
Let us taketon the both sides of (.). Then we have
(n+ )HB(nν,k)(x)
–
Thus, by (.), we obtain the following theorem.
Now, we observe that
whereBnare the ordinary Bernoulli numbers which are defined by the generating function to be
Therefore, by (.) and (.), we obtain the following theorem.
Theorem For n≥,we have
Now, we compute
in two different ways. On the one hand,
=
On the other hand,
Therefore, by (.) and (.), we obtain the following theorem.
Theorem For n≥,we have
Let us consider the following two Sheffer sequences:
HB(nν,k)(x)∼
Let us assume that
Then, by (.) and (.), we get
Therefore, by (.) and (.), we obtain the following theorem.
Theorem For n,r∈Z≥,we have generating function to be
Let us assume that
Therefore, by (.) and (.), we obtain the following theorem.
Theorem For n,r∈Z≥,we have
HB(nν,k)(x) = ( –λ)r
n
m=
n m
r
l=
r l
(–λ)r–lHB(nν–,km)(l)
Hm(r)(x|λ).
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 Korea government (MOE) (No. 2012R1A1A2003786).
Received: 23 September 2013 Accepted: 7 November 2013 Published:27 Nov 2013 References
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Cite this article as:Kim and Kim:Hermite and poly-Bernoulli mixed-type polynomials.Advances in Difference Equations
