Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2012, Article ID 927953, 11 pages
doi:10.1155/2012/927953
Research Article
Some Formulae of Products of
the Apostol-Bernoulli and Apostol-Euler
Polynomials
Yuan He1 and Chunping Wang1
1
Faculty of Science, Kunming University of Science and Technology, Kunming 650500, China
Correspondence should be addressed to Yuan He, hyyhe@yahoo.com.cn
Received 17 May 2012; Revised 13 July 2012; Accepted 15 July 2012
Academic Editor: Cengiz Çinar
Copyright q 2012 Y. He and C. Wang. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
Some formulae of products of the Apostol-Bernoulli and Apostol-Euler polynomials are
established by applying the generating function methods and some summation transform
techniques, and various known results are derived as special cases.
1. Introduction
The classical Bernoulli polynomials Bn x and Euler polynomials En x are usually defined
by means of the following generating functions:
∞
tn
text
Bn x
t
n!
e − 1 n0
|t| < 2π,
∞
tn
2ext
En x
t
n!
e 1 n0
|t| < π.
1.1
In particular, Bn Bn 0 and En 2n En 1/2 are called the classical Bernoulli numbers
and Euler numbers, respectively. These numbers and polynomials play important roles in
many branches of mathematics such as combinatorics, number theory, special functions, and
analysis. Numerous interesting identities and congruences for them can be found in many
papers; see, for example, 1–4.
Some analogues of the classical Bernoulli and Euler polynomials are the ApostolBernoulli polynomials Bn x; λ and Apostol-Euler polynomials En x; μ. They were
2
Discrete Dynamics in Nature and Society
respectively introduced by Apostol 5 see also Srivastava 6 for a systematic study and
Luo 7, 8 as follows:
∞
tn
text
Bn x; λ
t
n!
λe − 1
n0
∞
2ext
tn
En x; λ
t
n!
λe 1
n0
|t| < 2π if λ 1; |t| < log λ otherwise ,
1.2
|t| < π if λ 1; |t| < log−λ otherwise .
1.3
Moreover, Bn λ Bn 0; λ and En λ 2n En 1/2; λ are called the Apostol-Bernoulli
numbers and Apostol-Euler numbers, respectively. Obviously Bn x; λ and En x; λ reduce
to Bn x and En x when λ 1. Some arithmetic properties for the Apostol-Bernoulli and
Apostol-Euler polynomials and numbers have been well investigated by many authors. For
example, in 1998, Srivastava and Todorov 9 gave the close formula for the Apostol-Bernoulli
polynomials in terms of the Gaussian hypergeometric function and the Stirling numbers
of the second kind. Following the work of Srivastava and Todorov, Luo 7 presented the
close formula for the Apostol-Euler polynomials in a similar technique. After that, Luo
10 obtained some multiplication formulas for the Apostol-Bernoulli and Apostol-Euler
polynomials. Further, Luo 11 showed the Fourier expansions for the Apostol-Bernoulli and
Apostol-Euler polynomials by applying the Lipschitz summation formula and derived some
explicit formulae at rational arguments for these polynomials in terms of the Hurwitz zeta
function.
In the present paper, we will further investigate the arithmetic properties of the
Apostol-Bernoulli and Apostol-Euler polynomials and establish some formulae of products
of the Apostol-Bernoulli and Apostol-Euler polynomials by using the generating function
methods and some summation transform techniques. It turns out that various known results
are deduced as special cases.
2. The Restatement of the Results
For convenience, in this section we always denote by δ1,λ the Kronecker symbol given by
δ1,λ 0 or 1 according to λ / 1 or λ 1, and we also denote by maxa, b the maximum
number of the real numbers a, b and by x the maximum integer less than or equal to the
real number x. We now give the formula of products of the Apostol-Bernoulli polynomials in
the following way.
Theorem 2.1. Let m and n be any positive integers. Then,
1 Bnk y; λμ
Bm x; λBn y; μ n
−1 Bm−k y − x;
k
λ
nk
k0
n Bmk x; λμ
n
m
B
y − x; μ
k n−k
mk
k0
m!n!
1
m1
.
−1 δ1,λμ
Bmn y − x;
λ
m n!
m m
m−k
2.1
Discrete Dynamics in Nature and Society
3
Proof. Multiplying both sides of the identity
1
1
· v
u
λe − 1 μe − 1
λeu
1
1
v
u
uv
λe − 1 μe − 1 λμe
−1
2.2
by uvexuyv , we obtain
veyv
ue1x−yu
eyuv
vey−xv
exuv
uexu
·
λv
·
u
·
.
λeu − 1 μev − 1
λeu − 1 λμeuv − 1
μev − 1 λμeuv − 1
2.3
It follows from 2.3 that
uv
δ1,λμ
uv
e1x−yu ey−xv
λ u
v
λe − 1
μe − 1
δ1,λμ
veyv
ue1x−yu
eyuv
uexu
·
− λv
−
u
λe − 1 μev − 1
λeu − 1
λμeuv − 1 u v
δ1,λμ
exuv
vey−xv
−
.
−u v
μe − 1 λμeuv − 1 u v
2.4
By the Taylor theorem we have
∞
δ1,λμ exuv
∂n
−
λμeuv − 1 u v n0 ∂un
δ1,λμ
exu
−
u
λμe − 1
u
vn
.
n!
2.5
Since B0 x; λ 1 when λ 1 and B0 x; λ 0 when λ /
1 see e.g., 8, by 1.2 and 2.5 we
get
∞ ∞ B
δ1,λμ
exuv
um v n
nm1 x; λμ
−
·
·
.
λμeuv − 1 u v m0 n0 n m 1
m! n!
2.6
Putting 1.2 and 2.6 in 2.4, with the help of the Cauchy product, we derive
e1x−yu ey−xv
λ u
v
λe − 1
μe − 1
∞ m
∞ Bnk1 y; λμ um vn1
m
·
−λ
Bm−k 1 x − y; λ
k
nk1
m! n!
m0 n0 k0
∞ n
∞ Bmk1 x; λμ um1 vn
n
·
B
−
y − x; μ
k n−k
mk1
m! n!
m0 n0 k0
uv
δ1,λμ
uv
∞
∞ um vn
·
.
Bm x; λBn y; μ
m! n!
m0 n0
2.7
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Discrete Dynamics in Nature and Society
If we denote the left-hand side of 2.7 by M1 and
veyv
uexu
M2 λδ1,λ
− δ1,λμ δ1,μ
− δ1,λμ
λμev − 1
λμeu − 1
veyv
uexu
− δ1,μ
−
δ
,
− δ1,λ v
1,λ
μe − 1
λeu − 1
2.8
then applying 1.2 to 2.8, in light of 2.7, we have
∞ ∞
m 1
Bnk1 y; λμ
−λ m1
M1 M2 Bm1−k 1 x − y; λ
k
m 1 k0
nk1
m0 n0
n1 Bmk1 x; λμ
1 n1
−
Bn1−k y − x; μ
k
n 1 k0
mk1
Bm1 x; λ Bn1 y; μ um1 vn1
·
·
.
m1
n1
m!
n!
2.9
1 and
On the other hand, a simple calculation implies M1 M2 0 when λμ /
uv
M1 M2 δ1,λμ
uv
when λμ 1. Applying un of the summation, we obtain
n
δ1,1/λ
e1x−yu δ1,λ
ey−xv
−
−
λ u
λe − 1
u
v
1/λev − 1
n
k0 k u
2.10
vk −vn−k to 1.2, in view of changing the order
∞ ∞ B
e1x−yu δ1,λ
n
n1 1 x − y; λ
−
λ
λ u
u vk −vn−k
k
λe − 1
u
1!
n
k0 nk
∞ B
∞ n
n1 1 x − y; λ
λ
u vk1 −vn−k1
k
1
1!
n
k0 nk1
∞ B
−vn
n1 1 x − y; λ
·
.
λ
n1
n!
n0
2.11
Discrete Dynamics in Nature and Society
5
It follows from 1.2, 2.10, 2.11, and the symmetric relation for the Apostol-Bernoulli
polynomials λBn 1 − x; λ −1n Bn x; 1/λ for any nonnegative integer n see e.g., 8
that
M1 M2 uvδ1,λμ
∞
∞ −1
k Bn1
k0 nk1
y − x; 1/λ
n
u vk vn−k1
k1
n 1!
k y − x; 1/λ
k
n
δ1,λμ
um1 vn−m
−1
m
k
1
1!
n
m0
k0 nk1
∞
∞ ∞ Bn1 y − x; 1/λ
n
k
δ1,λμ
um1 vn−m
−1k
k
1
m
1!
n
m0 km nk1
∞
∞ k Bn1
2.12
y − x; 1/λ m1 n−m
u v
δ1,λμ
−1
n 1!
m0 nm1
∞
∞ um1 vn1
m m!n!Bnm2 y − x; 1/λ
·
.
·
δ1,λμ
−1
m!
n!
n m 2!
m0 n0
∞
∞ m Bn1
Thus, by equating 2.9 and 2.12 and then comparing the coefficients of um1 vn1 , we
complete the proof of Theorem 2.1 after applying the symmetric relation for the ApostolBernoulli polynomials.
It follows that we show some special cases of Theorem 2.1. By setting x y in
Theorem 2.1, we have the following.
Corollary 2.2. Let m and n be any positive integers. Then,
m 1 Bnk x; λμ
m
Bm x; λBn x; μ n
−1m−k Bm−k
k
λ
nk
k0
Bmk x; λμ
m
μ
B
k n−k
mk
k0
m!n!
1
Bmn
−1m1 δ1,λμ
.
λ
m n!
n n
2.13
It is well known that the classical Bernoulli numbers with odd subscripts obey B1 −1/2 and
B2n1 0 for any positive integer n see, e.g., 12. Setting λ μ 1 in Corollary 2.2, we
immediately obtain the familiar formula of products of the classical Bernoulli polynomials
due to Carlitz 13 and Nielsen 14 as follows.
6
Discrete Dynamics in Nature and Society
Corollary 2.3. Let m and n be any positive integers. Then,
Bm xBn x maxm/2,n/2
k0
Bmn−2k x
m
n
n
m
B2k
2k
2k
m n − 2k
m!n!
−1m1
Bmn .
m n!
2.14
Since the Apostol-Bernoulli polynomials Bn x; λ satisfy the difference equation
∂/∂xBn x; λ nBn−1 x; λ for any positive integer n see, e.g., 8, by substituting
x y for x in Theorem 2.1 and then taking differences with respect to y, we get the following
result after replacing x by x − y.
Corollary 2.4. Let m and n be any positive integers. Then,
m 1
1
1
m
Bn−1k y; λμ − Bm x; λBn−1 y; μ
−1m−k Bm−k y − x;
m k0 k
λ
m
n 1 1
n
−
Bn−k y − x; μ Bm−1k x; λμ Bn y; μ Bm−1 x; λ.
n k0 k
n
2.15
Setting x t and y 1 − t in Corollary 2.4, by λBn 1 − x; λ −1n Bn x; 1/λ for any
nonnegative integer n, we get the following.
Corollary 2.5. Let m and n be any positive integers. Then,
m 1
1
1
1
m
− Bm t; λBn−1 t;
−1k Bm−k 2t; λBn−1k t;
m k0 k
λμ
m
μ
n 1
1
1
1
n
Bm−1k t; λμ − B t;
Bm−1 t; λ.
−1k Bn−k 2t;
n k0 k
μ
n
μ
2.16
In particular, the case λ μ 1 in Corollary 2.5 gives the following generalization for
Woodcock’s identity on the classical Bernoulli numbers, see 15, 16,
Corollary 2.6. Let m and n be any positive integers. Then,
m 1
1
m
−1k Bm−k 2tBn−1k t − Bm tBn−1 t
m k0 k
m
n 1
1
n
−1k Bn−k 2tBm−1k t − Bn tBm−1 t.
n k0 k
n
2.17
We next present some mixed formulae of products of the Apostol-Bernoulli and ApostolEuler polynomials and numbers.
Discrete Dynamics in Nature and Society
7
Theorem 2.7. Let m and n be non-negative integers. Then,
m 1 Bnk1 y; λμ
m
m−k
Em x; λEn y; μ 2
−1 Em−k y − x;
k
λ
nk1
k0
n Bmk1 x; λμ
n
−2
E
y − x; μ
k n−k
mk1
k0
−1
m1
2.18
m!n!
1
Emn1 y − x;
2δ1,λμ
.
λ
m n 1!
Proof. Multiplying both sides of the identity
1
1
·
λeu 1 μev 1
λeu
1
1
−
λeu 1 μev 1 λμeuv − 1
2.19
by 2exuyv , we obtain
2exu
2eyv
2e1x−yu
eyuv
2ey−xv
exuv
1
· u
· v
λ
·
−
·
.
u
uv
v
2 λe 1 μe 1
λe 1 λμe
− 1 μe 1 λμeuv − 1
2.20
It follows from 2.20 that
δ1,λμ
uv
2e1x−yu 2ey−xv
λ
−
λeu 1
μev 1
δ1,λμ
2exu
2eyv
2e1x−yu
eyuv
1
·
−λ
−
· u
2 λe 1 μev 1
λeu 1
λμeuv − 1 u v
δ1,λμ
exuv
2ey−xv
−
.
μev 1 λμeuv − 1 u v
2.21
Applying 1.3 and 2.6 to 2.21, in view of the Cauchy product, we get
2e1x−yu 2ey−xv
λ
−
λeu 1
μev 1
∞ m ∞
Bnk1 y; λμ
1
m
Em x; λEn y; μ − λ
Em−k 1 x − y; λ
k
2
nk1
m0 n0
k0
n Bmk1 x; λμ um vn
n
·
.
y − x; μ
E
k n−k
mk1
m! n!
k0
δ1,λμ
uv
2.22
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Discrete Dynamics in Nature and Society
On the other hand, since the left-hand side of 2.22 vanishes when λμ /
1, it suffices to
consider the case λμ 1. Applying un nk0 nk u vk −vn−k to 1.3, in view of changing
the order of the summation, we have
λ
∞ E 1 x − y; λ ∞ 2e1x−yu
n
n
λ
u vk −vn−k
k
λeu 1
n!
k0 nk
∞ ∞ E 1 x − y; λ n
n
λ
u vk1 −vn−k1
k1
n!
k0 nk1
λ
2.23
∞
−vn
.
En 1 x − y; λ
n!
n0
It follows from 1.3, 2.23, and the symmetric relation for the Apostol-Euler polynomials
λEn 1 − x; λ −1n En x; 1/λ for any non-negative integer n see, e.g., 7 that
δ1,λμ
uv
2e1x−yu 2ey−xv
λ
−
λeu 1
μev 1
δ1,λμ
∞
∞ −1
k1 En
y − x; 1/λ
n
u vk vn−k1
k1
n!
k0 nk1
k y − x; 1/λ
n
k
δ1,λμ
um vn−m1
−1
k
1
m
n!
m0
k0 nk1
∞
∞ ∞ En y − x; 1/λ
n
k
δ1,λμ
um vn−m1
−1k1
k
1
m
n!
m0 km nk1
∞
∞ k1 En
2.24
y − x; 1/λ m n−m1
u v
δ1,λμ
−1
n!
m0 nm1
∞
∞ um v n
m1 m!n!Emn1 y − x; 1/λ
·
.
δ1,λμ
·
−1
m! n!
m n 1!
m0 n0
∞
∞ m1 En
Thus, by equating 2.22 and 2.24 and then comparing the coefficients of um vn , we
complete the proof of Theorem 2.7 after applying the symmetric relation for the ApostolEuler polynomials.
Next, we give some special cases of Theorem 2.7. By setting x y in Theorem 2.7, we
have the following.
Discrete Dynamics in Nature and Society
9
Corollary 2.8. Let m and n be non-negative integers. Then,
m 1 Bnk1 x; λμ
m
m−k
Em x; λEn x; μ 2
−1 Em−k 0;
k
λ
nk1
k0
n Bmk1 x; λμ
n
−2
E
0; μ
k n−k
mk1
k0
−1m1 2δ1,λμ
2.25
m!n!
1
.
Emn1 0;
λ
m n 1!
Since the classical Euler polynomials En x at zero arguments satisfy E0 0 1, E2n 0 0,
and E2n−1 0 1 − 22n B2n /n for any positive integer n see, e.g., 12, by setting λ μ 1
in Corollary 2.8, we obtain the following.
Corollary 2.9. Let m and n be non-negative integers. Then,
Em xEn x − 2
1 − 22k B2k
m
n
2k − 1
2k − 1
k
maxm1/2,n1/2
k1
Bnn2−2k x
2m!n!
−1m1
×
Km,n ,
m n 2 − 2k
m n 1!
2.26
where Km,n 21 − 2mn2 Bmn2 /m n 2 when m n ≡ 0 (mod 2) and Km,n 0 otherwise.
Theorem 2.10. Let m be non-negative integer and n positive integer. Then,
m n
1
m
Em x; λBn y; μ Enk−1 y; λμ
−1m−k Em−k y − x;
k
2 k0
λ
n n
k0
k
2.27
Bn−k y − x; μ Emk x; λμ .
Proof. Multiplying both sides of the identity
1
1
·
λeu 1 μev − 1
λeu
1
1
λeu 1 μev − 1 λμeuv 1
2.28
by 2vexuyv , we obtain
veyv
λv 2e1x−yu
2eyuv
vey−xv
2exuv
2exu
· v
·
·
·
.
u
u
uv
v
λe 1 μe − 1
2
λe 1 λμe
1 μe − 1 λμeuv 1
2.29
10
Discrete Dynamics in Nature and Society
By 1.3 and the Taylor theorem, we have
∞ ∞
um v n
2exuv
·
.
Enm x; λμ
uv
λμe
1 m0 n0
m! n!
2.30
Applying 1.2, 1.3, and 2.30 to 2.29, we get
∞ ∞
um vn
·
Em x; λBn y; μ
m! n!
m0 n0
∞ m
∞ um vn1
λ
m
·
Em−k 1 x − y; λ Enk y; λμ
2 m0 n0 k0 k
m! n!
∞ n
∞ um v n
n
·
,
y − x; μ Emk x; λμ
B
k n−k
m! n!
m0 n0 k0
2.31
which means
∞ E
∞ um vn1
m x; λBn1 y; μ
·
·
n1
m! n!
m0 n0
∞ m ∞
λ
m
Em−k 1 x − y; λ Enk y; λμ
k
2 k0
m0 n0
2.32
n1 um vn1
1 n1
·
.
Bn1−k y − x; μ Emk x; λμ
k
n 1 k0
m! n!
Thus, by comparing the coefficients of um vn1 in 2.32, we conclude the proof of
Theorem 2.10 after applying the symmetric relation for the Apostol-Euler polynomials.
Obviously, by setting x y in Theorem 2.10, we have the following.
Corollary 2.11. Let m be non-negative integer and n positive integer. Then,
m 1
n
m
m−k
Em x; λBn x; μ Enk−1 x; λμ
−1 Em−k 0;
2 k0 k
λ
n n
k0
k
Bn−k μ Emk x; λμ .
2.33
Since B1 −1/2, E0 0 1, E2n 0 0, and E2n−1 0 1 − 22n B2n /n for any positive integer
n, by setting λ μ 1 in Corollary 2.11, we obtain the following.
Discrete Dynamics in Nature and Society
11
Corollary 2.12. Let m be non-negative integer and n positive integer. Then,
Em xBn x maxm1/2,n/2
k1
22k − 1
m
n
n
B2k Emn−2k x
2k − 1
2k
2k
2.34
Emn x.
Remark 2.13. For the equivalent forms of Corollaries 2.9 and 2.12, the interested readers may
consult 14.
Acknowledgments
The authors are very grateful to anonymous referees for helpful comments on the previous
version of this work. They express their gratitude to Professor Wenpeng Zhang who provided
them with some suggestions. This paper is supported by the National Natural Science
Foundation of China Grant no. 10671194.
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International Journal of
Advances in
Combinatorics
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Mathematical Physics
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Volume 2014
Journal of
Complex Analysis
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Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
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Stochastic Analysis
Abstract and
Applied Analysis
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International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
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Applied Mathematics
Journal of
Function Spaces
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Volume 2014
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Volume 2014
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Volume 2014
Optimization
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Volume 2014
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Volume 2014