R E S E A R C H
Open Access
A note on higher-order Bernoulli polynomials
Dae San Kim
1and Taekyun Kim
2**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
LetPn={p(x)∈Q[x]|degp(x)≤n}be the (n+ 1)-dimensional vector space overQ.
From the property of the basisB(r)0,B(r)1,. . .,B(r)n for the spacePn, we derive some
interesting identities of higher-order Bernoulli polynomials.
1 Introduction
LetN={, , , . . .}andZ+=N∪ {}. For a fixedr∈Z+, thenth Bernoulli polynomials are
defined by the generating function to be
t et–
r
ext=eB(r)(x)t=
∞
n=
B(nr)(x)tn
n!
see [–] ()
with the usual convention about replacing (B(r)(x))nbyB(r)
n (x). In the special case,x= ,
B(nr)() =B(nr)are called thenth Bernoulli numbers of orderr. From (), we note that
B(nr)(x) = n
k=
n k
B(kr)xn–k= n
k=
n k
B(nr–)kxk
=
n+···+nr+nr+=n
n n, . . . ,nr,nr+
Bn· · ·Bnrx
nr+. ()
Thus, by () we get the Euler-type sums of products of Bernoulli numbers as follows:
B(nr)= n+···+nr=n
n n, . . . ,nr
BnBn· · ·Bnr
see [–]. ()
By () and (), we see thatB(nr)(x) is a monic polynomial of degreenwith coefficients inQ. From (), we note that
B(nr)(x)= d
dxB
(r)
n(x) =nB
(r)
n–(x)
see [–] ()
and
B(nr)(x+ ) –B(nr)(x) =nB(nr––)(x). ()
Let denote the space of real-valued differential functions on (–∞,∞) =R. Now, we define three linear operatorsI,,Donas follows:
If(x) =
x+
x
f(t)dt, f(x) =f(x+ ) –f(x), Df(x) =f(x). ()
Then we see that (i)DI=ID=, (ii)I=I, (iii)D=D.
Let Pn={p(x)∈Q(x)|degp(x)≤n} be the (n+ )-dimensional vector space overQ. Probably,{,x, . . . ,xn}is the most natural basis for this space. But{B(r)
(x),B (r) (x),B
(r) (x),
. . . ,B(nr)(x)}is also a good basis for the spacePnfor our purpose of arithmetical and com-binatorial applications.
Letp(x)∈Pn. Thenp(x) can be generated byB(r)(x),B (r) (x),B
(r)
(x), . . . ,B (r)
n (x) as follows:
p(x) = n
k=
akB(kr)(x).
In this paper, we develop methods for uniquely determining ak from the information
of p(x). From those methods, we derive some interesting identities of higher-order
Bernoulli polynomials.
2 Higher-order Bernoulli polynomials
Forr= , by (), we getB()n =xn(n∈Z+). Letp(x)∈Pn. For a fixedr∈Z+,p(x) can be generated byB(r)(x),B
(r) (x),B
(r)
(x), . . . ,B (r)
n (x) as follows:
p(x) = n
k=
akB(kr)(x). ()
From () and (), we can derive the following identities:
IB(r)
n (x) =
x+
x
B(r)
n(x)dx
=
n+
Bn(r+) (x+ ) –Bn(r+) (x). ()
By () and (), we get
IB(nr)(x) = n+
n+ B
(r–)
n (x) =B(nr–)(x). ()
It is easy to show that
B(nr)(x) =B(nr)(x+ ) –B(nr)(x) =nBn(r––)(x), ()
and
By () and (), we get
Irp(x) = n
k=
akB()k (x) = n
k=
akxk. ()
From () and (), we note that
DkIrp(x) = n
l=k
al
l! (l–k)!x
l–k. ()
Thus, by () we get
DkIrp() =k!ak. ()
Hence, from () we have
ak=
DkIrp()
k! . ()
Case . Letr>n. Thenr>kfor allk= , , , . . . ,n. By (), we get
ak=
k!D
kIkIr–kp() =
k!(DI)
kIr–kp()
=
k!
kIr–kp() =
k! k
j=
k j
(–)k–jIr–kp(j). ()
Case . Assume thatr≤n.
(i) For≤k≤r, by () we get
ak=
k! k
j=
(–)k–j
k j
Ir–kp(j). ()
(ii) Forr≤k≤n, by () we see that
ak=
k!D
k–rDrIrp() =
k!D
k–r(DI)rp() =
k!D
k–rrp()
=
k!
rDk–rp() =
k! r
j=
r j
(–)r–jDk–rp(j). ()
Therefore, by (), (), () and (), we obtain the following theorem.
Theorem
(a) Forr>n,we have
p(x) = n
k=
k
j=
k!(–) k–j
k j
(b) Forr≤n,we have
p(x) = r–
k=
k
j=
k!(–) k–j
k j
Ir–kp(j) B(kr)(x)
+ n
k=r
r
j=
k!(–)
r–jDk–rp(j) B(r)
k (x).
Let us take p(x) =xn ∈ P
n. Then xn can be expressed as a linear combination of
B(r),B(r), . . . ,B(nr). Forr>n, we have
Ir–kxn= n! (n+r–k)!
r–k
l=
(–)r–k–l
r–k l
(x+l)n+r–k. ()
Therefore, by Theorem and (), we obtain the following corollary.
Corollary For n,r∈Z+with r>n,we have
xn= n
k=
k
j=
r–k
l=
(–)r–j–l n!
k
j
r–k
l
k!(n+r–k)!(j+l) n+r–k
B(kr)(x).
Let us assume thatr,n∈Z+withr≤n. Observe that
Dk–rxn=n(n– )· · ·(n–k+r+ )xn–k+r= n! (n–k+r)!x
n–k+r. ()
Thus, by Theorem and (), we obtain the following corollary.
Corollary For n,r∈Z+with r≤n,we have
xn= r–
k=
k
j=
r–k
l=
(–)r–j–l n!
k
j
r–k
l
k!(n+r–k)!(j+l) n+r–k
B(kr)(x)
+ n
k=r
r
j=
(–)r–j n! r
j
k!(n+r–k)!j n+r–k
B(kr)(x).
Let us takep(x) =Bn(s)(x)∈Pn (s∈Z+). Thenp(x) can be generated by{B(r)(x),B (r) (x),
. . . ,B(nr)(x)}as follows:
B(ns)(x) = n
k=
akB(kr)(x). ()
Forr>n, we have
Ir–kB(ns)(x) =B(ns–r+k)(x). ()
Theorem For r,n,s∈Z+with r>n,we have
B(ns)(x) = n
k=
k
j=
(–)k–j
k!
k j
B(ns+k–r)(j)
B(kr)(x).
In particular, forr=s, we have
Bn(r)(x) = B(r)(x) + B(r)(x) +· · ·+ B(nr–) (x) + B(nr)(x)
= n
k=
k
j=
(–)k–j
k!
k j
B(nk)(j)
B(kr)(x). ()
By comparing coefficients on the both sides of (), we get
k
j=
(–)k–j
k!
k j
B(nk)(j) =δkn, for ≤k≤n. ()
Therefore, by (), we obtain the following corollary.
Corollary
(a) Forn,k∈Z+with≤k≤n– ,we have
k
j=
(–)k–j
k j
B(nk)(j) = .
(b) In particular,k=n,we get
n
j=
(–)n–j
n j
B(nn)(j) =n!.
Let us assume thatr≤nin (). Then we have
Dk–rB(ns)(x) =n(n– )· · ·(n–k+r+ )Bn(s+)r–k(x) = n! (n–k+r!)B
(s)
n+r–k(x). ()
Therefore, by Theorem , () and (), we obtain the following theorem.
Theorem For r,n∈Z+with r≤n,we have
B(ns)(x) = n–
k=
k
j=
(–)k–j
k!
k j
B(ns+k–r)(j)
B(kr)(x)
+ n
k=r
r
j=
(–)r–j n! r
j
k!(n+r–k)!B
(s)
n+r–k(j)
B(kr)(x).
Assume thatr,n∈Z+withr>n.
By (), we get
Ir–kE(ns)(x) =
(n+ )· · ·(n+r–k) r–k
l=
(–)r–k–l
r–k l
E(ns+)r–k(x+l)
= n!
(n+r–k)! r–k
l=
(–)r–k–l
r–k l
E(ns+)r–k(x+l). ()
Therefore, by Theorem and (), we obtain the following theorem.
Theorem For r,n∈Z+with r>n,we have
E(ns)(x) = n
k=
k
j=
r–k
l=
(–)r–j–l n!
k
j
r–k
l
k!(n+r–k)!E
(s)
n+r–k(j+l)
B(kr)(x).
Forr,n∈Z+withr≤n, we have
Dk–rE(ns)(x) =n(n– )· · ·(n–k+r+ )En(s–)k+r(x). ()
By Theorem and (), we obtain the following theorem.
Theorem For r,n∈Z+with r≤n,we have
E(ns)(x) = r–
k=
k
j=
r–k
l=
(–)r–j–l n!
k
j
r–k
l
k!(n+r–k)!E
(s)
n+r–k(j+l)
B(kr)(x)
+ n
k=r
r
j=
(–)r–j n! r
k
k!(n+r–k)!E
(s)
n+r–k(j)
B(kr)(x).
Remarks (a) Forr≤, by () we get
Irxn= n! (n+r)!
r
j=
r j
(–)r–j(x+j)n+r=n+r r
r! r
j=
(–)r–j
r j
(x+j)n+r.
Thus, forx= , we have
Irxn|x
==
n! (n+r)!
r
j=
r j
(–)r–jjn+r= n+r
r
r!
rn+r=S(n+r,r) n+r
r
, ()
whereS(n,r) is the Stirling number of the second kind. (b) Assume
n
k=
αkxk= n
k=
ApplyingIton both sides (t≥), we get
n
k=
akB(kr–t)(x) = n
k=
αkItxk= n
k=
αk α+t t
t! t
j=
(–)(t–j)
t j
(x+j)k+t. ()
From () and (), we have
n
k=
akB(kr–t)= n
k=
αk α+t t
S(k+t,t).
Remark Let us define two operatorsd,d˜as follows:
d=e–D=
∞
n=
(–)n
n! D
n, d˜=eD=
∞
n=
Dn
n!. ()
From (), we note that
˜
dxn= n
l=
n l
xn–l= (x+ )n,
dxn= n
l=
n l
(–)lxn–l= (x– )n.
()
Thus, by () and (), we get
˜
dB(nr)(x) =B(nr)(x+ ), dB(nr)(x) =B(nr)(x– ), ()
and
˜
dE(nr)(x) =E(nr)(x+ ), dE(nr)(x) =E(nr)(x– ). ()
3 Further remarks
For any r,r, . . . ,rn ∈ Z+, {B(r)(x),B (r)
(x), . . . ,B (rn)
(x)} forms a basis for Pn. Let r=
max{ri|i= , , , . . . ,n}. Letp(x)∈Pn. Thenp(x) can be expressed as a linear combina-tion ofB(r)
(x),B (r)
(x), . . . ,B (rn)
n (x) as follows:
p(x) =aB(r)(x) +aB(r)(x) +· · ·+anB(nrn)(x) = n
l=
alB(lrl)(x). ()
Thus, by () and (), we get
Irp(x) = n
l=
alIrB(lrl)(x)
= n
l=
alIr–rlIrlB(lrl)(x) = n
l=
alIr–rlB()l (x)
= n
l=
Now, for eachk= , , , . . . ,n, by () we get
DkIrp(x) = n
l=
alDkIr–rlxl= n
l=
alIr–rl
Dkxl
= n
l=k
alIr–rl
l! (l–k)!x
l–k
= n
l=k
all! (l–k)!I
r–rlxl–k. ()
Let us takex= in (). Then, by () and (), we get
DkIrp() = n
l=k
l!al (l–k)!×
S(l–k+r–rl,r–rl) l–k+r–rl
r–rl
= n
l=k
al(r–rl)!l!
(l–k)! S(l–k+r–rl,r–rl). ()
Case . Forr>n, we have
DkIrp() =DkIkIr–kp() = (DI)kIr–kp() =kIr–kp()
= k
j=
(–)k–j
k j
Ir–kp(j). ()
Case . Letr≤n.
(i) For≤k<r, we have
DkIrp() = k
j=
(–)k–j
k j
Ir–kp(j). ()
(ii) Forr≤k≤n, we have
DkIrp() =Dk–rDrIrp() =Dk–r(DI)rp() =Dk–rrp()
=rDk–rp() = r
j=
(–)r–j
r j
Dk–rp(j). ()
Thus, by (), (), () and (), we can determinea,a,a, . . . ,an.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both 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 paper is supported in part by the Research Grant of Kwangwoon University in 2013.
References
1. Abramowitz, M, Stegun, IA: Handbook of Mathematical Functions. Natl. Bur. of Standards, Washington (1964) 2. Araci, S, Acikgoz, M: A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials.
Adv. Stud. Contemp. Math - Jang’jun Math. Soc.22(3), 399-406 (2012)
3. Bayad, A, Kim, T: Identities involving values of Bernstein,q-Bernoulli, andq-Euler polynomials. Russ. J. Math. Phys. 18(2), 133-143 (2011)
4. Gould, HW: Explicit formulas for Bernoulli numbers. Am. Math. Mon.79, 44-51 (1972)
5. Kim, T: Symmetryp-adic invariant integral onZpfor Bernoulli and Euler polynomials. J. Differ. Equ. Appl.14(12),
1267-1277 (2008)
6. Kim, DS, Kim, T, Kim, YH, Dolgy, DV: A note on Eulerian polynomials associated with Bernoulli and Euler numbers and polynomials. Adv. Stud. Contemp. Math - Jang’jun Math. Soc.22(3), 379-389 (2012)
7. Kim, DS, Kim, T: Euler basis, identities, and their applications. Int. J. Math. Math. Sci.2012, Article ID 343981 (2012). doi:10.1155/2012/343981
8. Kim, DS, Kim, T: Bernoulli basis and the product of several Bernoulli polynomials. Int. J. Math. Math. Sci.2012, Article ID 463659 (2012). doi:10.1155/2012/463659
9. Kim, T: Sums of products ofq-Bernoulli numbers. Arch. Math.76(3), 190-195 (2001)
10. Kim, T, Adiga, C: Sums of products of generalized Bernoulli numbers. Int. Math. J.5(1), 1-7 (2004)
11. Kim, T: Symmetry of power sum polynomials and multivariate fermionicp-adic invariant integral onZp. Russ. J. Math.
Phys.16(1), 93-96 (2009)
12. Kim, T: Identities involving Frobenius-Euler polynomials arising from non-linear differential equations. J. Number Theory132(12), 2854-2865 (2012)
13. Kim, T: Some formulae for theq-Bernstein polynomials andq-deformed binomial distributions. J. Comput. Anal. Appl. 14(5), 917-933 (2012)
14. Petojevi´c, A: New sums of products of Bernoulli numbers. Integral Transforms Spec. Funct.19(1-2), 105-114 (2008) 15. Rim, S-H, Jeong, J: On the modifiedqEuler numbers of higher order with weight. Adv. Stud. Contemp. Math
-Jang’jun Math. Soc.22(1), 93-98 (2012)
16. Ryoo, CS: Some relations between twistedqEuler numbers and Bernstein polynomials. Adv. Stud. Contemp. Math -Jang’jun Math. Soc.21(2), 217-223 (2011)
17. 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)
doi:10.1186/1029-242X-2013-111
