R E S E A R C H
Open Access
Higher-order Frobenius-Euler 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 higher-order Frobenius-Euler polynomials, associated with poly-Bernoulli polynomials, which are derived from polylogarithmic function. These polynomials are called higher-order Frobenius-Euler and poly-Bernoulli mixed-type polynomials. The purpose of this paper is to give various identities of those polynomials arising from umbral calculus.
1 Introduction
Forλ∈Cwithλ= , the Frobenius-Euler polynomials of orderα(α∈R) are defined by the generating function to be
–λ
et–λ
α
ext=
∞
n= H(α)
n (x|λ) tn
n! (see [–]). (.)
Whenx= ,Hn(α)(λ) =Hn(α)(|λ) are called the Frobenius-Euler numbers of orderα. As is well known, the Bernoulli polynomials of orderαare defined by the generating function to be
t et–
α
ext=
∞
n=
B(α) n (x)
tn
n! (see [–]). (.)
Whenx= ,B(nα)=B(nα)(x) is called thenth Bernoulli number of orderα. In the special case,α= ,B()n (x) =Bn(x) is called thenth Bernoulli polynomial. Whenx= ,Bn=Bn() is called thenth ordinary Bernoulli number. Finally, we recall that the Euler polynomials of orderαare given by
et+
α
ext=
∞
n= En(α)(x)t
n
n! (see [–]). (.)
Whenx= ,E(nα)=En(α)() is called thenth Euler number of orderα. In the special case,
α= ,E()n (x) =En(x) is called thenth ordinary Euler polynomial. The classical polyloga-rithmic functionLik(x) is defined by
Lik(x) =
∞
n= xn
nk (k∈Z) (see []). (.)
As is known, poly-Bernoulli polynomials are defined by the generating function to be
Lik( –e–t) –e–t e
xt=
∞
n= B(k)n (x)t
n
n! (cf.[]). (.)
LetCbe the complex number field, and letFbe the set of all formal power series in the variabletoverCwith
F=
f(t) =
∞
k= ak
k!t ka
k∈C
. (.)
Now, we use the notationP=C[x]. In this paper,P∗will be denoted by the vector space of all linear functionals onP. Let us assume thatL|p(x)be the action of the linear functional Lon the polynomialp(x), and we remind 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. The formal power series
f(t) =
∞
k= ak
k!t
k∈F (.)
defines a linear functional onPby setting
f(t)|xn =an, for alln≥ (see [, ]). (.)
From (.) and (.), we note that
tk|xn =n!δn,k (see [, ]), (.)
whereδn,kis the Kronecker symbol.
Let us considerfL(t) =∞k=L|k!xntk. Then we see thatfL(t)|xn=L|xn, and soL=fL(t) as linear functionals. The mapL →fL(t) is a vector space isomorphism fromP∗ontoF. Henceforth, F will denote both the algebra of formal power series intand the vector space of all linear functionals onP, and so an elementf(t) ofF will be thought of as both a formal power series and a linear functional (see []). We shall callFthe umbral algebra. The umbral calculus is the study of umbral algebra. The ordero(f(t)) of a nonzero power seriesf(t) is the smallest integerk, for which the coefficient oftkdoes not vanish. A seriesf(t) is called a delta series ifo(f(t)) = , and an invertible series ifo(f(t)) = . Let f(t),g(t)∈F. Then we have
f(t)g(t)|p(x) =f(t)|g(t)p(x) =g(t)|f(t)p(x) (see []). (.)
For f(t),g(t)∈F with o(f(t)) = , o(g(t)) = , there exists a unique sequence Sn(x) (degSn(x) =n) such thatg(t)f(t)k|Sn(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)) (see [, ]). Letf(t)∈Fandp(t)∈P. Then we have
f(t) =
∞
k=
f(t)|xk t k
k!, p(x) =
∞
k=
tk|p(x)x k
From (.), we note that
p(k)() =tk|p(x) =|p(k)(x). (.)
By (.), we get
tkp(x) =p(k)(x) =d kp(x)
dxk (see [, ]). (.)
From (.), we easily derive the following equation
eytp(x) =p(x+y), eyt|p(x) =p(y). (.)
Forp(x)∈P,f(t)∈F, it is known that
f(t)|xp(x) =∂tf(t)|p(x) =
f(t)|p(x) (see []). (.)
LetSn(x)∼(g(t),f(t)). Then we have
g(f¯(x))e
yf¯(t)=
∞
n= Sn(y)t
n
n! for ally∈C, (.)
wheref¯(t) is the compositional inverse off(t) withf¯(f(t)) =t, and
f(t)Sn(x) =nSn–(x) (see [, ]). (.)
The Stirling number of the second kind is defined by the generating function to be
et– m=m!
∞
l=m S(l,m)
tm
m! (m∈Z≥). (.)
ForSn(x)∼(g(t),t), it is well known that
Sn+(x) =
x–g
(t)
g(t)
Sn(x) (n≥) (see [, ]). (.)
LetSn(x)∼(g(t),f(t)),rn(x)∼(h(t),l(t)). Then we have
Sn(x) = n
m=
Cn,mrm(x), (.)
where
Cn,m= m!
h(¯f(t)) g(f¯(t))l
¯
f(t)mxn
(see [, ]). (.)
2 Higher-order Frobenius-Euler polynomials, associated poly-Bernoulli polynomials
Let us consider the polynomialsTn(r,k)(x|λ), called higher-order Frobenius-Euler and poly-Bernoulli mixed-type polynomials, as follows:
–λ
et–λ
r
Lik( –e–t) –e–t e
xt=
∞
n=
Tn(r,k)(x|λ)t n
n!, (.)
whereλ∈Cwithλ= ,r,k∈Z.
Whenx= , Tn(r,k)(λ) =Tn(r,k)(|λ) is called thenth higher-order Frobenius-Euler and poly-Bernoulli mixed type number.
From (.) and (.), we note that
Tn(r,k)(x|λ)∼
gr,k(t) =
et–λ –λ
r
–e–t Lik( –e–t),t
. (.)
By (.) and (.), we get
tTn(r,k)(x|λ) =nTn–(r,k)(x|λ). (.)
From (.), we can easily derive the following equation
Tn(r,k)(x|λ) = n
l=
n l
Hn–l(r)(λ)B(k)l (x)
= n
l=
n l
Hn–l(r)(x|λ)B(k)l . (.)
By (.) and (.), we get
Tn(r,k)(x|λ) = gr,k(t)x
n=
–λ
et–λ
r
Lik( –e–t) –e–t x
n. (.)
In [], it is known that
Lik( –e–t) –e–t x
n= n
m= (m+ )k
m
j= (–)j
m j
(x–j)n. (.)
Thus, by (.) and (.), we get
Tn(r,k)(x|λ) =
–λ
et–λ
r
Lik( –e–t) –e–t x
n
=
∞
m= (m+ )k
m
j= (–)j
m j
–λ
et–λ
r (x–j)n
= n
m= (m+ )k
m
j= (–)j
m j
By (.), we easily see that
Therefore, by (.) and (.), we obtain the following theorem.
Theorem . For r,k∈Z,n≥,we have
Therefore, by (.) and (.), we obtain the following theorem.
Now, we note that
gr,k (t) gr,k(t)=
loggr,k(t)
=rloget–λ–rlog( –λ) +log –e–t–logLik –et
=r+ rλ etλ+
t et–
Lik( –e–t) –Lik–( –e–t) tLik( –e–t)
. (.)
By (.) and (.), we get
Tn+(r,k)(x|λ) =xTn(r,k)(x|λ) –rTn(r,k)(x|λ) – rλ –λ
–λ
et–λ
r+
Lik( –e–t) –e–t x
n
–
–λ
et–λ
r
Lik( –e–t) –Lik–( –e–t) t( –e–t)
t et–
xn
= (x–r)Tn(r,k)(x|λ) – rλ –λT
(r+,k) n (x|λ)
– n
l=
n l
Bn–l
–λ
et–λ
r
Lik( –e–t) –Lik–( –e–t) t( –e–t) x
l. (.)
It is easy to show that
Lik( –e–t) –Lik–( –e–t)
–e–t =
–e–t
∞
n=
( –e–t)n nk –
( –e–t)n nk–
=
–e–t
k –
–e–t k–
+· · ·
=
k –
k–
t+· · ·. (.)
For any delta seriesf(t), we have
f(t) t x
n=f(t) n+ x
n+. (.)
Thus, by (.), (.) and (.), we get
Tn+(r,k)(x|λ) = (x–r)Tn(r,k)(x|λ) – rλ –λT
(r+,k) n (x|λ)
– n
l=
n l
Bn–l l+
–λ
et–λ
r
Lik( –e–t) –Lik–( –e–t)
–e–t x
l+
= (x–r)Tn(r,k)(x|λ) – rλ –λT
(r+,k) n (x|λ)
– n
l=
n
l
l+ Bn–l
Tl+(r,k)(x|λ) –Tl+(r,k–)(x|λ)
= (x–r)Tn(r,k)(x|λ) – rλ –λT
– n+
n+
l=
n+ l
Bn+–lTl(r,k)(x|λ) –Tl(r,k–)(x|λ)
= (x–r)Tn(r,k)(x|λ) – rλ –λT
(r+,k) n (x|λ)
–
n+ n+
l=
n+ l
Bn+–lTl(r,k)(x|λ) –Tl(r,k–)(x|λ)
= (x–r)Tn(r,k)(x|λ) – rλ –λT
(r+,k) n (x|λ)
–
n+ n+
l=
n+ l
BlTn+–l(r,k) (x|λ) –Tn+–l(r,k–)(x|λ). (.)
Therefore, by (.), we obtain the following theorem.
Theorem . For r,k∈Z,n∈Z≥,we have
Tn+(r,k)(x|λ) = (x–r)Tn(r,k)(x|λ) – rλ –λT
(r+,k) n (x|λ)
–
n+ n+
l=
n+ l
BlTn+–l(r,k) (x|λ) –Tn+–l(r,k–)(x|λ).
Remark Ifr= , then we have
∞
n= B(k)n (x)t
n
n!=
Lik( –e–t) ( –e–t) e
xt=
∞
n=
Tn(,k)(x|λ)t n
n!. (.)
Thus, by (.), we getB(k)n (x) =Tn(,k)(x|λ). From (.), we have
txTn(r,k)(x|λ) =t
x n
l=
n l
Hn–l(r)(λ)B(k)l (x)
= n
l=
n l
Hn–l(r)(λ)lxB(k)l–(x) +B(k)l (x)
=nx n–
l=
n– l
Hn––l(r) (λ)B(k)l (x) + n
l=
n l
Hn–l(r)(λ)B(k)l (x)
=nxTn–(r,k)(x|λ) +Tn(r,k)(x|λ). (.)
Applyington both sides of Theorem ., we get
(n+ )Tn(r,k)(x|λ)
=nxTn–(r,k)(x|λ) +Tn(r,k)(x|λ) –rnTn–(r,k)(x|λ) – rnλ –λT
(r+,k) n– (x|λ)
–
n+ n+
l=
n+ l
Thus, by (.), we have
Therefore, by (.), we obtain the following theorem.
Theorem . For r,k∈Z,n∈Zwith n≥,we have
Therefore, by (.), we obtain the following theorem.
On the one hand,
On the other hand, we get
Therefore, by (.) and (.), we obtain the following theorem.
Now, we consider the following two Sheffer sequences:
Therefore, by (.) and (.), we obtain the following theorem.
Theorem . For r,k∈Z,s∈Z≥,we have By the same method, we get
From (.) and (.), we note that
Let us assume that
Tn(r,k)(x|λ) =
Therefore, by (.) and (.), we obtain the following theorem.
Theorem . For r,k∈Z,s∈Z≥,we have
It is known that
= n–m
l=
S(l+m,m) (l+m)! (n)m+l
–λ
et–λ
r
Lik( –e–t) –e–t x
n–m–l
= n–m
l=
n l+m
S(l+m,m)Tn–m–l(r,k) (λ). (.)
Therefore, by (.) and (.), we obtain the following theorem.
Theorem . For r,k∈Z,we have
Tn(r,k)(x|λ) = n
m=
n–m
l=
n l+m
S(l+m,m)Tn–m–l(r,k) (λ)
(x)m.
Finally, we consider the following two Sheffer sequences:
Tn(r,k)(x|λ)∼
et–λ
–λ r
–e–t Lik( –e–t)
,t
,
x[n]∼, –e–t,
(.)
wherex[n]=x(x+ )· · ·(x+n– ). Let us assume that
Tn(r,k)(x|λ) = n
m=
Cn,mx[m]. (.)
Then, by (.) and (.), we get
Cn,m= m!
–λ
et–λ
r
Lik( –e–t) –e–t
–e–tmxn
=
∞
l=
(–)lS(l+m,m) (l+m)!
–λ
et–λ
r
Lik( –e–t) –e–t
tm+lxn
= n–m
l=
(–)lS(l+m,m) (l+m)! (n)m+l
–λ
et–λ
r
Lik( –e–t) –e–t x
n–m–l
= n–m
l= (–)l
n l+m
S(l+m,m)Tn–m–l(r,k) (λ). (.)
Therefore, by (.) and (.), we obtain the following theorem.
Theorem . For r,k∈Z,n≥,we have
Tn(r,k)(x|λ) = n
m=
n–m
l= (–)l
n l+m
S(l+m,m)Tn–m–l(r,k) (λ)
x[m].
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: 11 July 2013 Accepted: 6 August 2013 Published: 20 August 2013 References
1. Araci, S, Acikgoz, M: A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials. Adv. Stud. Contemp. Math.22(3), 399-406 (2012)
2. Kim, T: An identity of the symmetry for the Frobenius-Euler polynomials associated with the fermionicp-adic invariantq-integrals onZp. Rocky Mt. J. Math.41(1), 239-247 (2011)
3. Kim, T: Identities involving Frobenius-Euler polynomials arising from non-linear differential equations. J. Number Theory132(1), 2854-2865 (2012)
4. Ryoo, C: A note on the Frobenius-Euler polynomials. Proc. Jangjeon Math. Soc.14(4), 495-501 (2011)
5. Ryoo, CS, Agarwal, RP: Exploring the multiple Changheeq-Bernoulli polynomials. Int. J. Comput. Math.82(4), 483-493 (2005)
6. Kim, DS, Kim, T, Kim, YH, Lee, SH: Some arithmetic properties of Bernoulli and Euler numbers. Adv. Stud. Contemp. Math.22(4), 467-480 (2012)
7. Kim, DS, Kim, T: Poly-Bernoulli polynomials arising from umbral calculus (communicated)
8. Kim, T: Power series and asymptotic series associated with theq-analog of the two-variablep-adicL-function. Russ. J. Math. Phys.12(2), 186-196 (2005)
9. 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(2), 135-160 (2009)
10. Ding, D, Yang, J: Some identities related to the Apostol-Euler and Apostol-Bernoulli polynomials. Adv. Stud. Contemp. Math.20(1), 7-21 (2010)
11. Kim, T, Choi, J: A note on the product of Frobenius-Euler polynomials arising from thep-adic integral onZp. Adv. Stud. Contemp. Math.22(2), 215-223 (2012)
12. Kurt, B, Simsek, Y: On the generalized Apostol-type Frobenius-Euler polynomials. Adv. Differ. Equ.2013, 1 (2013) 13. Simsek, Y, Yurekli, O, Kurt, V: On interpolation functions of the twisted generalized Frobenius-Euler numbers. Adv.
Stud. Contemp. Math.15(2), 187-194 (2007)
14. Roman, S: The Umbral Calculus. Pure and Applied Mathematics, vol. 111. Academic Press, New York (1984) 15. Roman, S, Rota, G-C: The umbral calculus. Adv. Math.27(2), 95-188 (1978)
doi:10.1186/1687-1847-2013-251
