R E S E A R C H
Open Access
A note on the higher-order Frobenius-Euler
polynomials and Sheffer sequences
Dae San Kim
1, Taekyun Kim
2*, Sang-Hun Lee
3and Seog-Hoon Rim
4*Correspondence:
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 investigate some properties of polynomials related to Sheffer sequences. Finally, we derive some identities of higher-order Frobenius-Euler polynomials.
1 Introduction
Letλ(= )∈C. The higher-order Frobenius-Euler polynomials are defined by the gener-ating function to be
–λ et–λ
α
ext=eH(α)(x|λ)t=
∞
n=
Hn(α)(x|λ)t n
n! (see [–]) ()
with the usual convention about replacing (H(α)(x|λ))nbyH(α)
n (x|λ). In the special case,
x= ,Hn(α)(|λ) =Hn(α)(λ) are called thenth Frobenius-Euler numbers of orderα(∈R). From () we have
Hn(α)(x|λ) = n
l=
n
l
Hn–l(α)(λ)xl= n
l=
n l
Hn–l(α)(λ)xn–l (see []). ()
By () we get
∂ ∂xH
(α)
n (x|λ) =nH (α)
n–(x|λ), Hn()(x|λ) =xn forn∈Z+. ()
It is not difficult to show that
Hn(α)(x+ |λ) –λHn(α)(x|λ) = ( –λ)Hn(α–)(x|λ) (see [–]). ()
Let us define theλ-difference operatorλas follows:
λf(x) =f(x+ ) –λf(x). ()
From () we can derive the following equation:
As is well known, the Stirling numbersS(l,n) of the second kind are defined by the gener-ating function to be
et– n=n!
Now, we consider theλ-analogue of the Stirling numbers of the second kind as follows:
LetFbe the set of all formal power series in the variabletoverCwith
F=
f(t) =
∞
n=
ak
k!t ka
k∈C
.
Pindicates the algebra of polynomials in the variablexoverC, andP*is the vector space of all linear functionals onP(see [, ]). In [],L|p(x)denotes the action of the linear functionalLon a polynomialp(x), and we remind that the vector space structure onP*is defined by
L+M|p(x)=L|p(x)+M|p(x),
cL|p(x)=cL|p(x),
wherecis a complex constant. The formal power series
f(t) =
∞
k=
ak
k!t
k∈F ()
defines a linear functional onPby setting
f(t)|xn=an for alln∈Z+(see []). ()
From () and (), we have
tk|xn=n!δn,k (see [, ]). ()
LetfL(t) =∞k=L|k!xktk. From () we have
fL(t)|xn=L|xn for alln∈Z+. ()
By () we getL=fL(t). It is known in [] that the mapL→fL(t) is a vector space isomor-phism fromP*ontoF. Henceforth,Fwill denote 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 will callFthe umbral algebra. The umbral calculus is the study of umbral algebra (see [, ]).
The orderO(f(t)) of the nonzero power seriesf(t) is the smallest integerkfor which the coefficient oftkdoes not vanish. A seriesf(t) hasO(f(t)) = is called a delta series and a seriesf(t) hasO(f(t)) = is called an invertible series (see [, ]). By () and (), we get eyt|xn=yn, and soeyt|p(x)=p(y) (see [, ]). Forf(t)∈Fandp(x)∈P, we have
f(t) =
∞
k=
f(t)|xk
k! t
k, p(x) =
∞
k=
tk|p(x)
k! x
Letf(t),f(t), . . . ,fn(t)∈F. Then we see that
f(t)f(t)· · ·fn(t)|xn
=
i+···+im=n
n i, . . . ,im
f(t)|xi· · ·fm(t)|xim, ()
where i n
,...,im
=i n!
!···im! (see [, ]).
Forf(t),g(t)∈Fandp(x)∈P, it is easy to show that
f(t)g(t)|p(x)=f(t)|g(t)p(x)=g(t)|f(t)p(x) (see []). ()
From (), we can derive the following equation:
p(k)() =tk|p(x) and |p(k)(x)=p(k)(). ()
By () we get
tkp(x) =p(k)(x) =d kp(x)
dxk (see [, ]). ()
Thus, from () we have
eytp(x) =p(x+y). ()
Let Sn(x) be polynomials in the variablexwith degree n, and letf(t) be a delta series andg(t) be an invertible series. Then there exists a unique sequenceSn(x) of polynomials withg(t)f(t)k|Sn(x)=n!δ
n,k(n,k≥), whereδn,kis the Kronecker symbol. The sequence
Sn(x) is called the Sheffer sequence for (g(t),f(t)), which is denoted bySn(x)∼(g(t),f(t)). IfSn(x)∼(,f(t)), thenSn(x) is called the associated sequence forf(t). IfSn(x)∼(g(t),t), thenSn(x) is called the Appell sequence forg(t) (see [, ]). Forp(x)∈P, the following equations ()-() are known in [, ]:
eyt–
t p(x)
=
y
p(u)du, ()
f(t)|xp(x)=∂tf(t)|p(x)
=f(t)|p(x), ()
and
eyt– |p(x)=p(y) –p(). ()
ForSn(x)∼(g(t),f(t)), we have
h(t) =
∞
k=
h(t)|Sk(x)
k! g(t)f(t)
k, h(t)∈F, ()
p(x) =
∞
k=
g(t)f(t)k|p(x)
f(t)Sn(x) =nSn–(x), f(t)|p(αx)=f(αt)|p(x), ()
g(f¯(t))e y¯f(t)=
∞
k=
Sk(y)
k! t
k for ally∈C, ()
wheref¯(t) is the compositional inverse off(t).
Sn(x+y) = n
k=
n k
Pk(y)Sn–k(x) = n
k=
n k
Pk(x)Sn–k(y), ()
wherePk(y) =g(t)Sk(y)∼(,f(t)) (see [, ]).
In contrast to the higher-order Euler polynomials, the more general higher-order Frobenius-Euler polynomials have never been studied in the context of umbral algebra and umbral calculus.
In this paper, we investigate some properties of polynomials related to Sheffer se-quences. Finally, we derive some identities of higher-order Frobenius-Euler polynomials.
2 Associated sequences
Letpn(x)∼(,f(t)) andqn(x)∼(,g(t)). Then, forn≥, we note that
qn(x) =x
f(t)
g(t)
n
x–pn(x) (see []). ()
Let us takef(t) =eat– (a= ). Then we see thatf(t) =aeat,¯f(t) =a–log(t+ ). From (), we can derive the associated sequencepn(x) forf(t) =eat– as follows:
pn(y) =eyt|pn(x)=eyf¯(t)|xn=eyalog(t+)|xn
=(t+ )ay|xn= ∞
k=
y a
k
tk|xn
= n
k=
y a
k
n!δn,k=
y a
n
n! =
y a
n
, ()
where (a)n=a(a– )· · ·(a–n+ ) =n–i=(a–i). Therefore, by () we obtain, forn∈Z+,
pn(x) =
x a
n
∼ ,eat– .
We get the following:
pn+(x) =x f(t)–pn(x) =a–xe–atpn(x)
=
x a
x–a
a
n =
x a
From (), we can derive the equation
Therefore, by () and (), we obtain the following theorem.
Lemma For n,k≥,we have
As is well known, thenth Frobenius-Euler polynomials are defined by the generating function to be
It is easy to show thatxn∼(,t) (see Eq. ()). Thus, from () we have
3 Frobenius-Euler polynomials of order
α
From () and (), we note thatThen we have
Therefore, by () we obtain the following proposition.
Thus, we have
Therefore, by () we obtain the following corollary.
Corollary For n≥,we have
Theorem Forα∈Nand n≥,we have
4 Further remark
Thus, by () we get
From () and (), we can derive the following equation:
(x+ )n=
Therefore, by () and (), we obtain the following lemma.
Lemma For≤m≤n,we have
Theorem For n≥,we have
Hn(x|λ) =Hn(λ) + n
k= n
l=
n l
Hn–l(λ)S(l,k)(x)k
=Hn(λ) + n
k= n
l= k
m=
n
l
Hn–l(λ)S(l,k)S(k,m)xm.
From the recurrence formula of the Appell sequence, we note that
Hn+(α)(x|λ) =
x–α e
t
et–λ
Hn(α)(x|λ) =xHn(α)(x|λ) – αe t
et–λH (α) n (x|λ)
=xHn(α)(x|λ) – α ( –λ)
–λ et–λH
(α) n (x+ )
=xHn(α)(x|λ) – α ( –λ)H
(α+)
n (x+ ). ()
Therefore, by () we obtain the following theorem.
Theorem For n≥,we have
Hn+(α)(x|λ) =xHn(α)(x|λ) – α ( –λ)H
(α+) n (x+ ).
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.3Division of General Education, Kwangwoon University, Seoul,
139-701, Republic of Korea.4Department of Mathematics Education, Kyungpook National University, Taegu, 702-701,
Republic of Korea.
Acknowledgements
Dedicated to Professor Hari M Srivastava.
The authors express their sincere gratitude to the referees for their valuable suggestions and comments. This paper is supported in part by the Research Grant of Kwangwoon University in 2013.
Received: 9 December 2012 Accepted: 6 February 2013 Published: 21 February 2013
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doi:10.1186/1687-1847-2013-41
Cite this article as:Kim et al.:A note on the higher-order Frobenius-Euler polynomials and Sheffer sequences.
