We give some explicit relationships between the Apostol-Euler polynomials and generalized Hurwitz-Lerch Zeta function and obtain some series representations of the Apostol-Euler polynomials of higher order in terms of the generalized Hurwitz-Lerch Zeta function. Several interesting special cases are also shown

Qiu-Ming Luo

CiteSeerX

Applicable Analysis and Discrete Mathematics
available online at http://pefmath.etf.bg.ac.yu
Appl. Anal. Discrete Math. x (xxxx), xxx–xxx. doi:10.2298/AADMxxxxxxxx
SOME FORMULAS FOR APOSTOL-EULER POLYNOMIALS
ASSOCIATED WITH HURWITZ ZETA FUNCTION AT
RATIONAL ARGUMENTS
Qiu-Ming Luo
We give some explicit relationships between the Apostol-Euler polynomials
and generalized Hurwitz-Lerch Zeta function and obtain some series repre-
sentations of the Apostol-Euler polynomials of higher order in terms of the
generalized Hurwitz-Lerch Zeta function. Several interesting special cases are
also shown.
1. Introduction
Throughout this paper, we always make use of the following notations: N = {1, 2, 3, . . .}
denotes the set of natural numbers, N0 = {0, 1, 2, 3, . . .} denotes the set of nonnegative integers,
Z−0 = {0,−1,−2,−3, . . .} denotes the set of nonpositive integers, Z denotes the set of integers,
R denotes the set of real numbers, C denotes the set of complex numbers.
The generalized Bernoulli polynomials B(α)n (x) and Euler polynomials E
(α)
n (x) of order α
(real or complex) are usually defined by means of the following generating functions (see, for
details, [1], [5], [13], [15]):
(1.1)
(
z
ez − 1
)α
exz =
∞∑
n=0
B(α)n (x)
zn
n!
(|z| < 2π)
and
(1.2)
(
2
ez + 1
)α
exz =
∞∑
n=0
E(α)n (x)
zn
n!
(|z| < π) .
Obviously, the classical Bernoulli polynomials Bn(x) and Euler polynomials En(x) are defined
by
(1.3) Bn (x) := B(1)n (x) and En (x) := E
(1)
n (x) (n ∈ N0) ,
respectively. The classical Bernoulli numbers Bn and Euler numbers En are defined by
Bn := Bn (0) and En := 2nEn
(
1
2
)
(n ∈ N0) ,(1.4)
respectively.
2000 Mathematics Subject Classification: Primary 11M35; Secondary 11B68.
Keywords and Phrases: Apostol-Bernoulli polynomials, Apostol-Euler polynomials, Hurwitz Zeta function, Hurwitz-
Lerch Zeta function, Lerch’s functional equation.
1
Applicable Analysis and Discrete Mathematics - Accepted manuscript
Received: March 2, 2009.,
Revised: May 25, 2009.
Please cite this article as: Qiu-Ming Luo: SOME FORMULAS FOR APOSTOL-EULER POLYNOMIALS ASSOCIATED 
WITH HURWITZ ZETA FUNCTION AT RATIONAL ARGUMENTS,
Appl. Anal. Discrete Math. (to appear)
2 Qiu-Ming Luo
Some interesting analogues of the classical Bernoulli polynomials and numbers were first
investigated by Apostol [2, p. 165, Eq. (3.1)] and (more recently) by Srivastava [14, p. 83-84].
We begin by recalling Apostol’s definitions as follows:
Definition 1.1 (Apostol [2]; see also Srivastava [14]). The Apostol-Bernoulli polynomials
Bn(x;λ) in x are defined by means of the generating function:
zexz
λez − 1 =
∞∑
n=0
Bn(x;λ)z
n
n!
(1.5)
(|z| < 2π when λ = 1; |z| < |log λ| when λ 6= 1)
with, of course,
(1.6) Bn(x) = Bn(x; 1) and Bn (λ) := Bn (0;λ) ,
where Bn (λ) denotes the so-called Apostol-Bernoulli numbers (In fact, it is a function in λ).
Recently, Luo and Srivastava extended further the Apostol-Bernoulli and Apostol-Euler
polynomials and their generalizations as follows:
Definition 1.2 (cf. Luo and Srivastava [10, 12]). The Apostol-Bernoulli polynomials
B(α)n (x;λ) of order α are defined by means of the generating function
(
z
λez − 1
)α
exz =
∞∑
n=0
B(α)n (x;λ)
zn
n!
(1.7)
(|z| < 2π when λ = 1; |z| < |log λ| when λ 6= 1)
with, of course,
B(α)n (x) = B(α)n (x; 1) and B(α)n (λ) := B(α)n (0;λ) ,
Bn (x;λ) := B(1)n (x;λ) and Bn (λ) := Bn (0;λ) ,
(1.8)
where Bn (λ), B(α)n (λ) and Bn (x;λ) denote the so-called Apostol-Bernoulli numbers, Apostol-
Bernoulli numbers of order α and Apostol-Bernoulli polynomials respectively.
Definition 1.3 (cf. Luo [11]). The Apostol-Euler polynomials E(α)n (x;λ) of order α are
defined by means of the generating function
(1.9)
(
2
λez + 1
)α
exz =
∞∑
n=0
E(α)n (x;λ)
zn
n!
(|z| < |log(−λ)|) ,
with, of course,
E(α)n (x) = E(α)n (x; 1) and E(α)n (λ) := 2nE(α)n
(α
2
;λ
)
,
En (x;λ) := E(1)n (x;λ) and En (λ) := 2nEn
(α
2
;λ
)
,
(1.10)
where En (λ), E(α)n (λ) and En (x;λ) denote the so-called Apostol-Euler numbers, Apostol-Euler
numbers of order α and Apostol-Euler polynomials respectively.
The main object of the present paper is to give some explicit relationships between the
Apostol-Euler polynomials and generalized Hurwitz-Lerch Zeta function and to investigate some
series representations of the Apostol-Euler polynomials in terms of generalized Hurwitz-Lerch
Zeta function.
Some formulas for Apostol-Euler polynomials associated with Hurwitz Zeta function at rational arguments 3
2. Some explicit relationships between the Apostol-Euler polynomials and the
generalized Hurwitz-Lerch Zeta function
A family of the Hurwitz-Lerch Zeta function Φ(ρ,σ)µ,ν (z, s, a) defined by (see e.g. [9, p. 727,
Eq. (8)]
(2.1) Φ(ρ,σ)µ,ν (z, s, a) :=
∞∑
n=0
(µ)ρn
(ν)σn
zn
(n + a)s
(µ ∈ C; a, ν ∈ C \ Z−0 ; ρ, σ ∈ R+; ρ < σ when s, z ∈ C;
ρ = σ and s ∈ C when |z| < 1; ρ = σ and <(s− µ + ν) > 1 when |z| = 1),
contains, as its special cases, not only the Hurwitz-Lerch Zeta function
(2.2) Φ(σ,σ)ν,ν (z, s, a) = Φ
(0,0)
µ,ν (z, s, a) = Φ(z, s, a) =
∞∑
n=0
zn
(n + a)s
and the Lipschitz-Lerch Zeta function (cf. [15, p. 122, Eq. 2.5 (11)]):
(2.3) φ(ξ, a, s) := Φ
(
e2πiξ, s, a
)
=
∞∑
n=0
e2nπiξ
(n + a)s
(
a ∈ C \ Z−0 ; <(s) > 0 when ξ ∈ R \Z; <(s) > 1 when ξ ∈ Z
)
,
but also the following generalized Hurwitz-Lerch Zeta functions introduced and studied earlier
by Goyal and Laddha [7, p. 100, Eq. (1.5)]
(2.4) Φ(1,1)µ,1 (z, s, a) = Φµ(z, s, a) :=
∞∑
n=0
(µ)n
n!
zn
(n + a)s
,
which, for convenience, are called the Goyal-Laddha-Hurwitz-Lerch Zeta function. Here the
symbol (a)k denotes the Pochhammer symbol or the shifted factorial defined, a ∈ C, by
(2.5) (a)k =
Γ(a + k)
Γ(a)
=
{
1 (k = 0)
a(a + 1) · · · (a + k − 1) (k ∈ N),
where Γ(x) is the usual Gamma function.
Recently, Garg et al. [6] obtained the following interesting formula:
(2.6) B(l)n (a;λ) = (−n)lΦl(λ, l − n, a) (n, l ∈ N; n = l; |λ| < 1; a ∈ C \ Z−0 ).
Clearly, we have
(2.7) Bn (a;λ) = −nΦ(λ, 1− n, a) (n ∈ N; |λ| 5 1; a ∈ C \ Z−0 ).
Below we give the following explicit relationships between the a family of Euler polynomials and
a family of Zeta function.
Theorem 2.1. For n ∈ N; − 1 < λ 5 1; α ∈ C; a ∈ C \ Z−0 , the following relationship
(2.8) E(α)n (a;λ) = 2αΦα(−λ,−n, a)
between the Apostol-Euler polynomials of higher order and the Goyal-Laddha-Hurwitz-Lerch Zeta
function.
4 Qiu-Ming Luo
Proof. By (1.9) and the generalized binomial theorem, yields
∞∑
n=0
E(α)n (a;λ)
zn
n!
=
(
2
λez + 1
)α
eaz = 2α
∞∑
k=0
(α)k
k!
(−λ)ke(k+a)z
=
∞∑
n=0
[
2α
∞∑
k=0
(α)k
k!
(−λ)k(k + a)n
]
zn
n!
=
∞∑
n=0
[
2α
∞∑
k=0
(α)k
k!
(−λ)k
(k + a)−n
]
zn
n!
.
(2.9)
Hence, the formula (2.8) follows. ¤
Corollary 2.2. For n ∈ N; − 1 < λ 5 1; a ∈ C \ Z−0 , the following relationship
(2.10) En (a;λ) = 2Φ(−λ,−n, a)
holds true between the Apostol-Euler polynomials and the Hurwitz-Lerch Zeta function.
It is well-known that the following relationship between the Bernoulli polynomials and the
Hurwitz Zeta function (see Apostol [3, p. 264, Theorem 12.13])
(2.11) Bn(a) = −nζ(1− n, a) (n ∈ N),
where ζ(s, a) denotes the Hurwitz Zeta function defined by
ζ(s, a) := Φ(1, s, a) =
∞∑
n=0
1
(n + a)s
(< (s) > 1; a ∈ C \ Z−0 ).
An alternating series version of the Hurwitz Zeta function is given as follows:
Definition 2.3. The L-function is defined by
(2.12) L(s, a) :=
∞∑
n=0
(−1)n
(n + a)s
(< (s) > 1; a ∈ C \ Z−0
)
.
In the same method, it is not difficult, we give a quasi formula of (2.11) as follows:
Theorem 2.4. For n ∈ N; a ∈ C \ Z−0 , the following relationship
(2.13) En (a) = 2L(−n, a)
holds true between the Euler polynomials and the L-function.
It is well-known that the following relationship between the Bernoulli numbers and the
Riemann Zeta function (see [3, p. 266, Theorem 12.16])
(2.14) Bn = −nζ(1− n) (n ∈ N),
where ζ(s) denotes the Riemann Zeta function defined by
ζ(s) := ζ(s, 1) =
∞∑
n=1
1
ns
.
An alternating series version of the Riemann Zeta function is given as follows:
Definition 2.5. For <(s) > 0, the l-function defined by
(2.15) l(s) :=
∞∑
n=1
(−1)n
ns
.
Similarly, we give an analogue of the formula (2.14) as follows:
Some formulas for Apostol-Euler polynomials associated with Hurwitz Zeta function at rational arguments 5
Theorem 2.6. For n ∈ N, the following relationship
(2.16) En = 2l(−n)
holds true between the Euler numbers and the l-function.
3. Explicit series representations for the Apostol-Euler polynomials of order α
It is not difficult, we make use of the elementary series identity:
(3.1)
∞∑
k=1
f (k) =
q∑
j=1
∞∑
k=0
f (qk + j) , (q ∈ N) ,
to the Hurwitz-Lerch Zeta function (2.2), yields that
(3.2) Φ(z, s, a) = q−s
q∑
j=1
Φ
(
zq, s,
a + j − 1
q
)
zj−1.
Obviously, a special case of (3.2) when
z = exp
(
2pπi
q
)
(p ∈ Z, q ∈ N)
is the following summation formula for the Lipschitz-Lerch Zeta function φ(ξ, a, s) defined by
(2.3):
Φ
(
exp
(
2pπi
q
)
, s, a
)
= φ
(
p
q
, a, s
)
= q−s
q∑
j=1
ζ
(
s,
a + j − 1
q
)
exp
(
2(j − 1)pπi
q
)
,
(3.3)
in terms of the Hurwitz Zeta function ζ(s, a).
Theorem 3.1. For n, q ∈ N; p ∈ Z; ξ ∈ R, α ∈ C, the following formula of the Apostol-
Euler polynomials of order α
E(α)n
(
p
q
; e2πiξ
)
=
i 2α · n!
Γ(α)
∞∑
k=0
(
α− 1
k
)(
k − p
q
+ 1
)
α−k−1
k∑
j=0
(
k − 1
j − 1
)
(n + 1)jB
(k)
k−j
· (2πq)−n−j−1
{
q∑
r=1
ζ
(
n + j + 1,
2ξ + 2r − 1
2q
)
· exp
[(
n + j
2
− (2ξ + 2r − 1)p
2
)
πi
]
−
q∑
r=1
ζ
(
n + j + 1,
2r − 2ξ + 1
2q
)
· exp
[(
−n + j
2
+
(2r − 2ξ + 1)p
2
)
πi
]}
,
(3.4)
holds true in terms of the Hurwitz Zeta function.
6 Qiu-Ming Luo
Proof. We now rewrite the result of Lin et al. as follows (see [8, p. 823, Theorem]):
Φµ(z, s, a) =iz−aΓ(1− s)
∞∑
k=0
(k − a + 1)µ−k−1
k!Γ(µ− k)
k∑
j=0
(
k − 1
j − 1
)
(1− s)jB(k)k−j(2π)s−j−1
·
[
exp
(
−1
2
(s− j)πi
)
Φ
(
e−2πai, 1− s + j, log z
2πi
)
− exp
[(
2a +
1
2
(s− j)
)
πi
]
Φ
(
e2πai, 1− s + j, 1− log z
2πi
)]
(µ ∈ C).
(3.5)
Setting
z = −e2πiξ, a = p
q
and s 7→ −s, µ 7→ α
and by applying the series identity (3.3), we find that
Φα
(
−e2πiξ,−s, p
q
)
=
iΓ(s + 1)
Γ(α)
∞∑
k=0
(
α− 1
k
)(
k − p
q
+ 1
)
α−k−1
k∑
j=0
(
k − 1
j − 1
)
(s + 1)jB
(k)
k−j
· (2πq)−s−j−1
{
q∑
r=1
ζ
(
s + j + 1,
2ξ + 2r − 1
2q
)
· exp
[(
s + j
2
− (2ξ + 2r − 1)p
q
)
πi
]
−
q∑
r=1
ζ
(
s + j + 1,
2r − 2ξ + 1
2q
)
· exp
[(
−s + j
2
+
(2r − 2ξ + 1)p
q
)
πi
]}
(α ∈ C).
(3.6)
Taking s = n in (3.6) and noting that (2.8) of Theorem 2.1, of course, with
λ = e2πiξ and a =
p
q
,
we obtain the desire (3.4). This proof is complete. ¤
Theorem 3.2. For n, q, l ∈ N; p ∈ Z; ξ ∈ C, the following formula of the Apostol-Euler
polynomials of order l
E(l)n
(
p
q
; e2πiξ
)
=
i2l · n!
(l − 1)!
l−1∑
k=0
(
l − 1
k
)(
k − p
q
+ 1
)
l−k−1
k∑
j=0
(
k − 1
j − 1
)
(n + 1)jB
(k)
k−j
· (2πq)−n−j−1
{
q∑
r=1
ζ
(
n + j + 1,
2ξ + 2r − 1
2q
)
exp
[(
n + j
2
− (2ξ + 2r − 1)p
2
)
πi
]
−
q∑
r=1
ζ
(
n + j + 1,
2r − 2ξ + 1
2q
)
exp
[(
−n + j
2
+
(2r − 2ξ + 1)p
2
)
πi
]}
,
(3.7)
holds true in terms of the Hurwitz Zeta function.
Proof. Let α = l (l ∈ N) in (3.6) we may obtain the assertion (3.7). ¤
Some formulas for Apostol-Euler polynomials associated with Hurwitz Zeta function at rational arguments 7
Theorem 3.3. For n, q, l ∈ N; p ∈ Z; ξ ∈ C, the following formula of the Apostol-Euler
polynomials of order l
E(l)n
(
p
q
; e2πiξ
)
=− i(−2)
l · n!
(l − 1)!
l−1∑
k=0
(
l − 1
k
)
B
(l)
l−k−1q
−n−k−1
k∑
j=0
(
k
j
)(−n− 1
j
)
j!pk−j(2π)−n−j−1
·
{
q∑
r=1
ζ
(
n + j + 1,
2ξ + 2r − 1
2q
)
exp
[(
n + j
2
− (2ξ + 2r − 1)p
q
)
πi
]
−
q∑
r=1
ζ
(
n + j + 1,
2r − 2ξ + 1
2q
)
exp
[(
−n + j
2
+
(2r − 2ξ + 1)p
q
)
πi
]}
,
(3.8)
holds true in terms of the Hurwitz Zeta function.
Proof. Setting µ = m (m ∈ N) in (3.5), we obtain the following transformation formula:
Φm(z, s, a) =
iz−aΓ(1− s)
(m− 1)!
m−1∑
k=0
(
m− 1
k
)
B
(m)
m−k−1
·
k∑
j=0
(−1)m−k+j−1
(
k
j
)(
s− 1
j
)
j!(−a)k−j(2π)s−j−1
·
[
exp
(
1
2
(s− j)πi
)
Φ
(
e−2πai, 1− s + j, log z
2πi
)
− exp
[(
2a +
1
2
(s− j)
)
πi
]
Φ
(
e2πai, 1− s + j, 1− log z
2πi
)]
(m ∈ N).
(3.9)
Letting
z = −e2πiξ, a = p
q
and s 7→ −s, m 7→ l
and by applying the series identity (3.3), we obtain the following consequence
Φl
(
−e2πiξ,−s, p
q
)
=
i(−1)l−1Γ(s + 1)
(l − 1)!
l−1∑
k=0
(
l − 1
k
)
B
(l)
l−k−1q
−s−k−1
k∑
j=0
(
k
j
)(−s− 1
j
)
j!pk−j(2π)−s−j−1
·
{
q∑
r=1
ζ
(
s + j + 1,
2ξ + 2r − 1
2q
)
exp
[(
s + j
2
− (2ξ + 2r − 1)p
q
)
πi
]
−
q∑
r=1
ζ
(
s + j + 1,
2r − 2ξ + 1
2q
)
exp
[(
−s + j
2
+
(2r − 2ξ + 1)p
q
)
πi
]}
(l ∈ N),
(3.10)
Further taking s = n in (3.10) and noting that (2.8) of Theorem 2.1, of course, with
λ = e2πiξ and a =
p
q
(p ∈ Z; q ∈ N; ξ ∈ R).
Therefore, the formula (3.8) follows. This proof is complete. ¤
8 Qiu-Ming Luo
4. Further observations and consequences
Recently, Srivastava found the following elegant formula for Apostol-Bernoulli polynomials
Bn(x;λ) (see [14, p. 84, Eq. (4.6)]):
Bn
(
p
q
; e2πiξ
)
=− n!
(2qπ)n



q∑
j=1
ζ
(
n,
ξ + j − 1
q
)
exp
[(
n
2
− 2(ξ + j − 1)p
q
)
πi
]
+
q∑
j=1
ζ
(
n,
j − ξ
q
)
exp
[(
−n
2
+
2(j − ξ)p
q
)
πi
]}
(4.1)
(n ∈ N \ {1} ; p ∈ Z; q ∈ N; ξ ∈ R).
When ξ ∈ Z in (4.1), we can deduce a known result given earlier by Cvijović and Klinowski [4,
p. 1529, Theorem A]:
(4.2) Bn
(
p
q
)
= − 2 · n!
(2qπ)n
q∑
j=1
ζ
(
n,
j
q
)
cos
(
2jpπ
q
− nπ
2
)
(n ∈ N \ {1} ; p ∈ Z; q ∈ N) .
It follows that we set α = 1 in (3.4), or l = 1 in (3.7) and (3.8). Then we obtain the following
interesting formula for the Apostol-Euler polynomials En(x;λ).
Theorem 4.1. For n, q ∈ N; p ∈ Z; ξ ∈ R, the following formula of the Apostol-Euler
polynomials at rational arguments
En
(
p
q
; e2πiξ
)
=
2 · n!
(2qπ)n+1
{
q∑
j=1
ζ
(
n + 1,
2ξ + 2j − 1
2q
)
exp
[(
n + 1
2
− (2ξ + 2j − 1)p
q
)
πi
]
+
q∑
j=1
ζ
(
n + 1,
2j − 2ξ − 1
2q
)
exp
[(
−n + 1
2
+
(2j − 2ξ − 1)p
q
)
πi
]}
(4.3)
holds true in terms of the Hurwitz Zeta function.
A special case of formula (4.3) when ξ ∈ Z, is just a known result given earlier by Cvijović
and Klinowski:
Corollary 4.2 ([4, p. 1529, Theorem B]). For n, q ∈ N; p ∈ Z, the following formula of
the classical Euler polynomials
En
(
p
q
)
=
4 · n!
(2qπ)n+1
q∑
j=1
ζ
(
n + 1,
2j − 1
2q
)
sin
(
(2j − 1)pπ
q
− nπ
2
)
,(4.4)
holds true in terms of the Hurwitz Zeta function.
Acknowledgements
The author greatly appreciates to the referee for the valuable comments and suggestions.
The present investigation was supported in part by the PCSIRT Project of the Ministry of
Education of China under Grant IRT0621 and Innovation Program of Shanghai Municipal Edu-
cation Committee of China under Grant 08ZZ24 and Henan Innovation Project For University
Prominent Research Talents of China under Grant 2007KYCX0021.
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Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics
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Department of Mathematics, Jiaozuo University, Henan Jiaozuo 454003, People’s Republic of China.
Department of Mathematics, East China Normal University, Dongchuan Road 500, Shanghai 200241,
People’s Republic of China.
E–mail: luomath2007@163.com
9


SOME FORMULAS FOR APOSTOL-EULER POLYNOMIALS ASSOCIATED WITH HURWITZ ZETA FUNCTION AT RATIONAL ARGUMENTS

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SOME FORMULAS FOR APOSTOL-EULER POLYNOMIALS ASSOCIATED WITH HURWITZ ZETA FUNCTION AT RATIONAL ARGUMENTS

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