UDC 517.9

Q.-M. Luo (Chongqing Normal Univ., China)

q-APOSTOL – EULER POLYNOMIALS AND q-ALTERNATING SUMS*

q-ПОЛIНОМИ АПОСТОЛА – ЕЙЛЕРА ТА q-ЗНАКОЗМIННI СУМИ

We establish the basic properties and generating functions of the q-Apostol – Euler polynomials. We define q-alternating
sums and obtain q-extensions of some formulas in [Integral Transform. Spec. Funct. – 2009. – 20. – P. 377 – 391]. We also
deduce an explicit relationship between the q-Apostol – Euler polynomials and the q-Hurwitz – Lerch zeta-function.

Встановлено основнi властивостi та твiрнi функцiї q-полiномiв Апостола – Ейлера. Визначено q-знакозмiннi суми
та отримано q-продовження деяких формул з [Integral Transform. Spec. Funct. – 2009. – 20. – P. 377 – 391]. Виведено
також явне спiввiдношення мiж q-полiномами Апостола – Ейлера i q-дзета-функцiєю Хурвiца – Лерча.

1. Introduction and definitions. Throughout this paper, we always use the following notation:
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 q-shifted factorial are defined by

(a; q)0 = 1, (a; q)k = (1− a)(1− aq) . . . (1− aqk−1), k = 1, 2, . . . ,

(a; q)∞ = (1− a)(1− aq) . . . (1− aqk) . . . =
∞∏
k=0

(1− aqk).

The q-numbers are defined by [a]q =
1− qa

1− q
, q 6= 1.

The above q-standard notation can be found in Gasper [12, p. 7].

The classical Bernoulli polynomials and Euler polynomials are defined by means of the following
generating functions (see, e.g., [1, p. 804 – 806], [11] or [25, p. 25 – 32]):

zexz

ez − 1
=
∞∑
n=0

Bn(x)
zn

n!
, |z| < 2π, (1.1)

and

2exz

ez + 1
=

∞∑
n=0

En(x)
zn

n!
, |z| < π, (1.2)

respectively. Obviously, Bn := Bn(0) and En := En (0) are the Bernoulli numbers and Euler
numbers respectively.

Some interesting analogues of the classical Bernoulli polynomials were first investigated by
Apostol. We begin by recalling here Apostol’s definition as follows:

* This research was supported by Natural Science Foundation Project of Chongqing, China under Grant
CSTC2011JJA00024, Research Project of Science and Technology of Chongqing Education Commission, China under
Grant KJ120625, Fund of Chongqing Normal University, China under Grant 10XLR017 and 2011XLZ07 and the PhD
Program Scholarship Fund of ECNU 2009 of China under Grant 2009041.

c© Q.-M. LUO, 2013

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q-APOSTOL – EULER POLYNOMIALS AND q-ALTERNATING SUMS 1105

Definition 1.1 [2]. The Apostol – Bernoulli polynomials Bn(x;λ) are defined by means of the
generating function

zexz

λez − 1
=

∞∑
n=0

Bn(x;λ)
zn

n!
(1.3)

(|z| < 2π when λ = 1; |z| < |log λ| when λ 6= 1)

with, of course,

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 further extended the Euler polynomials based on the Apostol’s idea [2] as follows:
Definition 1.2 (cf. [18]). The Apostol – Euler polynomials En (x;λ) are defined by means of the

generating function

2exz

λez + 1
=
∞∑
n=0

En (x;λ)
zn

n!
, |z| < |log(−λ)| , (1.4)

with, of course,

En(x) = En(x; 1) and En (λ) := En (0;λ) , (1.5)

where En (λ) denote the so-called Apostol – Euler numbers.

Recently, M. Cenkci and M. Can [6] further defined the following q-extensions of the Apostol –
Bernoulli polynomials, i.e., the so-called q-Apostol – Bernoulli polynomials.

Definition 1.3. The q-Apostol – Bernoulli numbers Bn;q(λ) and polynomials Bn (x;λ) are de-
fined by means of the generating functions

Uλ;q(t) = −t
∞∑
n=0

λnqne[n]qt =
∞∑
n=0

Bn;q(λ)
tn

n!
(1.6)

and

Ux;λ;q(t) = −t
∞∑
n=0

λnqn+xe[n+x]qt =
∞∑
n=0

Bn;q(x;λ)
tn

n!
(1.7)

respectively.

Setting λ = 1 in (1.6) and (1.7), we obtain the corresponding Carlitz’s definitions for the q-
Bernoulli numbers Bn;q and q-Bernoulli polynomials Bn;q(x) respectively.

Obviously,

lim
q→1

Bn;q(x) = Bn(x), lim
q→1

Bn;q = Bn

and

lim
q→1
Bn;q(x;λ) = Bn(x;λ), lim

q→1
Bn;q(λ) = Bn(λ).

It follows that we define the following q-extensions of the Apostol – Euler numbers and polyno-
mials (see [3 – 6, 9]).

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1106 Q.-M. LUO

Definition 1.4. The q-Apostol – Euler numbers En;q(λ) and polynomials En (x;λ) are defined
by means of the generating functions

Vλ;q(t) = 2

∞∑
n=0

(−λ)nqne[n]qt =

∞∑
n=0

En;q(λ)
tn

n!
(1.8)

and

Vx;λ;q(t) = 2

∞∑
n=0

(−λ)nqn+xe[n+x]qt =

∞∑
n=0

En;q(x;λ)
tn

n!
(1.9)

respectively.

When λ = 1, then the above definitions (1.8) and (1.9) will become the corresponding definitions
of the q-Euler numbers En;q and q-Euler polynomials En;q(x).

Clearly,

lim
q→1

En;q(x) = En(x), lim
q→1

En;q = En

and

lim
q→1
En;q(x;λ) = En(x;λ), lim

q→1
En;q(λ) = En(λ).

There are numerous recent investigations on this subject by, among many other authors, Cenki et
al. [6 – 8], Choi et al. [9, 10], Kim [14 – 16], Luo and Srivastava [17 – 24], Ozden [26] and Simsek
[27 – 29].

The aim of the present paper is to investigate the basic properties, generating functions, Raabe’s
multiplication theorem and alternating sums for the q-Apostol – Euler polynomials and to obtain some
q-extensions of some formulas in [Integral Transform. Spec. Funct. – 2009. – 20. – P. 377 – 391]. We
also derive some interesting formulas and relationships between the q-Apostol – Euler polynomials,
the q-Apostol – Euler polynomials and q-Hurwitz – Lerch zeta function.

2. The properties of the q-Apostol – Euler polynomials. The following elementary properties of
the q-Apostol – Euler polynomials En;q (x;λ) are readily derived from (1.8) and (1.9). We, therefore,
choose to omit the details involved.

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q-APOSTOL – EULER POLYNOMIALS AND q-ALTERNATING SUMS 1107

Proposition 2.1 (the several values).

E0;q(x;λ) =
2

λq + 1
qx,

E1;q(x;λ) =
2

λq + 1
qx[x]q −

2λ

(λq + 1)(λq2 + 1)
q2x+1,

E0;q(λ) =
2

λq + 1
,

E1;q(λ) = − 2λq

(λq + 1)(λq2 + 1)
,

E2;q(λ) =
2λq(λq2 − 1)

(λq + 1)(λq2 + 1)(λq3 + 1)
,

E3;q(λ) = − 2λq(λ2q5 − 4λq2 + 1)

(λq + 1)(λq2 + 1)(λq3 + 1)(λq4 + 1)
.

(2.1)

Proposition 2.2. An expansion formula of q-Apostol – Euler polynomials

En;q(x;λ) =

n∑
k=0

(
n

k

)
Ek;q(λ)q(k+1)x[x]n−kq . (2.2)

Proposition 2.3 (difference equation).

λEn;q(x+ 1;λ) + En;q(x;λ) = 2qx[x]nq . (2.3)

Proposition 2.4 (differential relation).

∂

∂x
En;q(x;λ) = En;q(x;λ) log q + n

log q

q − 1
qxEn−1;q(x;λq). (2.4)

Proposition 2.5 (integral formula).

b∫
a

qxEn;q(x;λq) dx =
1− q
n+ 1

b∫
a

En+1;q(x;λ) dx+
q − 1

log q

En+1;q(b;λ)− En+1;q(a;λ)

n+ 1
. (2.5)

Proposition 2.6 (addition theorem).

En;q(x+ y;λ) =

n∑
k=0

(
n

k

)
Ek;q(x;λ)q(k+1)y[y]n−kq . (2.6)

Proposition 2.7 (theorem of complement).

En;q(1− x;λ) =
(−1)n

λqn
En;q−1(x;λ−1), (2.7)

En;q(1 + x;λ) =
(−1)n

λqn
En;q−1(−x;λ−1). (2.8)

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1108 Q.-M. LUO

Remark 2.1. If q → 1, then the formulas (2.1) – (2.8) become the corresponding formulas
for the Apostol – Euler polynomials (see [18, p. 918, 919], Eqs. (3) – (11) when α = 1). So the
above formulas are q-extensions of the corresponding formulas of the Apostol – Euler polynomials
respectively.

Remark 2.2. When λ = 1, then the formulas (2.1) – (2.8) become the corresponding formulas
for the q-Euler polynomials (see [3 – 5]).

3. The generating functions of q-Apostol – Euler polynomials. By (1.8) and (1.9) yields that

Vx;λ;q(t) = 2
∞∑
n=0

(−λ)nqn+xe[n+x]qt =

= 2e
t

1−q

∞∑
n=0

(−λ)nqn+xe
− qn+x

1−q
t

=

= 2e
t

1−q

∞∑
k=0

(−1)kq(k+1)x

(1− q)k
tk

k!

∞∑
n=0

(−λqk+1)n =

= 2e
t

1−q

∞∑
k=0

(−1)kq(k+1)x

1 + λqk+1

(
1

1− q

)k tk
k!
. (3.1)

Therefore, we obtain the generating function of En;q(x;λ) as follows:

Vx;λ;q(t) = 2e
t

1−q

∞∑
k=0

(−1)kq(k+1)x

1 + λqk+1

(
1

1− q

)k tk
k!

=
∞∑
n=0

En;q(x;λ)
tn

n!
. (3.2)

Clearly, setting x = 0 in (3.2) we have the generating function of En;q(λ):

Vλ;q(t) = 2e
t

1−q

∞∑
k=0

(−1)k

1 + λqk+1

(
1

1− q

)k tk
k!

=

∞∑
n=0

En;q(λ)
tn

n!
. (3.3)

Putting λ = 1 in (3.2) and (3.3), we deduce the generating function of En;q(x) and En;q

Vx;q(t) = 2e
t

1−q

∞∑
k=0

(−1)kq(k+1)x

1 + qk+1

(
1

1− q

)k tk
k!

=
∞∑
n=0

En;q(x)
tn

n!
(3.4)

and

Vq(t) = 2e
t

1−q

∞∑
k=0

(−1)k

1 + qk+1

(
1

1− q

)k tk
k!

=
∞∑
n=0

En;q
tn

n!
(3.5)

respectively.
It is not difficult, from (3.2) and (3.3) we get the following closed formulas:

En;q (x;λ) =
2

(1− q)n
n∑
k=0

(
n

k

)
(−1)kq(k+1)x

1 + λqk+1
(3.6)

and

En;q (λ) =
2

(1− q)n
n∑
k=0

(
n

k

)
(−1)k

1 + λqk+1
. (3.7)

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q-APOSTOL – EULER POLYNOMIALS AND q-ALTERNATING SUMS 1109

Remark 3.1. In the same way, we can also obtain the generating function of q-Apostol –
Bernoulli polynomials as follows:

Ux;λ;q(t) = −te
t

1−q

∞∑
k=0

(−1)kq(k+1)x

1− λqk+1

(
1

1− q

)k tk
k!

=

∞∑
n=0

Bn;q(x;λ)
tn

n!
. (3.8)

4. q-Raabe’s multiplication theorem, q-alternating sums and their applications.
Theorem 4.1 (q-Apostol-Raabe’s multiplication theorem). For m,n ∈ N, λ ∈ C, then we have

En;q(mx;λ) =


[m]nq

m−1∑
j=0

(−λ)jEn;qm
(
x+

j

m
;λm

)
, m is odd,

− 2

n+ 1
[m]nq

m−1∑
j=0

(−λ)jBn+1;qm

(
x+

j

m
;λm

)
, m is even.

(4.1)

Proof. If m is odd, we compute the following sum by (3.2):

∞∑
n=0

[m]nq

m−1∑
j=0

(−λ)jEn;qm
(
x+

j

m
;λm

) tn
n!

=

= 2e
t

1−q

∞∑
k=0

(−1)kq(k+1)mx

1 + (λqk+1)
m

(
1

1− q

)k tk
k!

m−1∑
j=0

(
−λqk+1

)j
=

= 2e
t

1−q

∞∑
k=0

(−1)kq(k+1)mx

1 + λqk+1

(
1

1− q

)k tk
k!

=

=
∞∑
n=0

En;q(mx;λ)
tn

n!
. (4.2)

Comparing the coefficients of
tn

n!
on the both sides of (4.2), we obtain the first formula of Theo-

rem 4.1.
If m is even, we calculate the following sum by (3.8) and (3.2):

∞∑
n=0

− 2

n+ 1
[m]nq

m−1∑
j=0

(−λ)jBn+1;qm

(
x+

j

m
;λm

) tn
n!

=

= 2e
t

1−q

∞∑
k=0

(−1)kq(k+1)mx

1− (λqk+1)
m

(
1

1− q

)k tk
k!

m−1∑
j=0

(
−λqk+1

)j
=

= 2e
t

1−q

∞∑
k=0

(−1)kq(k+1)mx

1 + λqk+1

(
1

1− q

)k tk
k!

=

=

∞∑
n=0

En;q(mx;λ)
tn

n!
. (4.3)

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1110 Q.-M. LUO

Comparing the coefficients of
tn

n!
on the both sides of (4.3), we obtain the second formula of Theo-

rem 4.1.
Theorem 4.1 is proved.

Clearly, the above formulas (4.1) of Theorem 4.1 are a q-extensions of the multiplication formulas in
[20, p. 386] (Eq. (43)).

Taking λ = 1 in (4.1), we obtain the following corollary.
Corollary 4.1 (q-Raabe’s multiplication theorem). For m,n ∈ N, then we have

En;q(mx) =


[m]nq

m−1∑
j=0

(−1)jEn;qm

(
x+

j

m

)
, m is odd,

− 2

n+ 1
[m]nq

m−1∑
j=0

(−1)jBn+1;qm

(
x+

j

m

)
, m is even.

(4.4)

Obviously, the above formulas (4.4) are a q-extensions of the classical Raabe’s multiplication theorem
of Euler polynomials in [20, p. 386] (Eq. (45)).

We now define the following alternating sums:

Zk;q(m;n;λ) =
m∑
j=1

(−1)j+1λjqj(n−k)[j]kq =

= λqn−k[1]kq − λ2q2(n−k)[2]kq + . . .+ (−1)m+1λmqm(n−k)[m]kq , (4.5)

Zk;q(m;n) =

m∑
j=1

(−1)j+1qj(n−k)[j]kq = qn−k[1]kq − q2(n−k)[2]kq + . . .+ (−1)m+1qm(n−k)[m]kq ,

(4.6)

Zk(m;λ) =
m∑
j=1

(−1)j+1λjjk = λ1k − λ22k + . . .+ (−1)m+1λmmk, (4.7)

Zk(m) =
m∑
j=1

(−1)j+1jk = 1k − 2k + . . .+ (−1)m+1mk, (4.8)

m,n, k ∈ N; n ≥ k; λ ∈ C,

which are called the q-λ-alternating sums, q-alternating sums, λ-alternating sums and alternating
sums respectively.

It is easy to obtain the following generating functions of Zk(m;λ) and Zk(m) respectively:
∞∑
k=0

Zk(m;λ)
xk

k!
=

m∑
j=1

(−1)j+1λjejx =
(−λ)m+1e(m+1)x + λex

λex + 1
, (4.9)

∞∑
k=0

Zk(m)
xk

k!
=

m∑
j=1

(−1)j+1ejx =
(−1)m+1e(m+1)x + ex

ex + 1
. (4.10)

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q-APOSTOL – EULER POLYNOMIALS AND q-ALTERNATING SUMS 1111

Theorem 4.2. Let m be odd. For m,n ∈ N;λ ∈ C, the recursive formula of q-Apostol – Euler
numbers

[m]nq En;qm(λm)− En;q(λ) =

n∑
k=0

(
n

k

)
[m]kqEk;qm(λm)Zn−k;q(m− 1;n+ 1;λ) (4.11)

holds true in terms of the q-λ-alternating sums defined by (4.5).

Proof. If m is odd, taking x = 0 in (4.1) we obtain

En;q(λ) = [m]nq

m−1∑
j=0

(−λ)jEn;qm
(
j

m
;λm

)
=

=
n∑
k=0

(
n

k

)
[m]kqEk;qm(λm)

m−1∑
j=0

(−λ)jq(k+1)j [j]n−kq =

= −
n∑
k=0

(
n

k

)
[m]kqEk;qm(λm)Zn−k;q(m− 1;n+ 1;λ) + [m]nq En;qm(λm). (4.12)

The formula (4.11) follows.
Theorem 4.2 is proved.
Clearly, the above formula (4.11) is an q-extension of the formula in [20, p. 389] (Eq. (60)).
Putting λ = 1 in (4.11), we obtain the following corollary.
Corollary 4.2. Let m be odd. For m,n ∈ N, the recursive formula of Apostol – Euler numbers

holds

[m]nqEn;qm − En;q =

n∑
k=0

(
n

k

)
[m]kqEk;qmZn−k;q(m− 1;n+ 1) (m is odd). (4.13)

Clearly, the above formula (4.13) is a q-extension of the formula in [20, p. 389] (Eq. (61)).
Theorem 4.3. Let m be even. For m,n ∈ N, λ ∈ C, the formula of q-Apostol – Euler numbers

[m]qEn;q(λ) +
2

n+ 1
[m]n+1

q Bn+1;qm(λm) =

=
2

n+ 1

n+1∑
k=0

(
n+ 1

k

)
[m]kqBk;qm(λm)Zn+1−k;q(m− 1;n+ 1;λ), (4.14)

holds true in terms of the q-λ-alternating sums defined by (4.5).

Proof. If m is even, setting x = 0 in (4.1) we have

En;q(λ) = − 2

n+ 1
[m]nq

m−1∑
j=0

(−λ)jBn+1;qm

(
j

m
;λm

)
=

=
2

n+ 1

n+1∑
k=0

(
n+ 1

k

)
[m]k−1q Bk;qm(λm)

m−1∑
j=0

(−1)j+1λjqkj [j]n+1−k
q =

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1112 Q.-M. LUO

=
2

n+ 1

n+1∑
k=0

(
n+ 1

k

)
[m]k−1q Bk;qm(λm)Zn−k+1;q(m− 1;n+ 1;λ)− 2

n+ 1
[m]nqBn+1;qm(λm).

(4.15)
The formula (4.3) follows.
Theorem 4.3 is proved.
Clearly, the above formula (4.3) is a q-extension of the formula (see [20, p. 390], Eq. (63) for

` = 1):

mEn(λ)+
2

n+ 1
mn+1Bn+1(λ

m) =
2

n+ 1

n+1∑
k=0

(
n+ 1

k

)
mkBk(λm)Zn+1−k(m−1;λ) (m is even),

(4.16)
where λ-alternating sums defined by (4.7).

Putting λ = 1 in (4.3), we have the following corollary.
Corollary 4.3. For m be even, m,n ∈ N; λ ∈ C, the formula of q-Euler numbers

[m]qEn;q +
2

n+ 1
[m]n+1

q Bn+1;qm =
2

n+ 1

n+1∑
k=0

(
n+ 1

k

)
[m]kqBk;qmZn+1−k;q(m−1;n+1), (4.17)

holds true in terms of the q-alternating sums defined by (4.6).
Clearly, the above formula (4.17) is a q-extension of the formula in [20, p. 390] (Eq. (64) for

` = 1).
Remark 4.1. Setting λ = 1 in (4.16) and noting that Z0(m− 1) = 1 for m even, we derive the

following interesting sum formula:
n∑
k=0

(
n+ 1

k

)
mkBkZn+1−k(m− 1) =

m(n+ 1)

2
En (n ∈ N; m is even). (4.18)

Applying the relation En =
2

n+ 1
(1− 2n+1)Bn+1 to (4.18), we find that

n∑
k=0

(
n+ 1

k

)
mkBkZn+1−k(m− 1) = m(1− 2n+1)Bn+1 (n ∈ N; m is even), (4.19)

which is just the formula of Howard (see [13, p. 167], Eq. (33)).

Remark 4.2. Separatting the odd and even terms in (4.19), and noting that B0 = 1, B1 =
1

2
,

B2n+1 = 0, n ∈ N, we have
n−1∑
k=1

(
2n

2k

)
m2kB2kZ2n−2k(m− 1) = m(1− 22n)B2n +mnZ2n−1(m− 1)− Z2n(m− 1) (4.20)

(n ∈ N; m is even)

and
n∑
k=1

(
2n+ 1

2k

)
m2kB2kZ2n−2k+1(m− 1) =

m(2n+ 1)

2
Z2n(m− 1)− Z2n+1(m− 1) (4.21)

(n ∈ N; m is even),

where the alternating sums Zk(m) defined by (4.8).

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q-APOSTOL – EULER POLYNOMIALS AND q-ALTERNATING SUMS 1113

Below we give the evaluations for the alternating sums (4.5), (4.6) given by (4.22) and (4.24)
respectively.

Theorem 4.4. For m, n ∈ N, λ ∈ C, the following formula of q-λ-alternating sums:

Zn;q(m;n+ 1;λ) =
m∑
j=0

(−1)j+1λjqj [j]nq =
(−λ)m+1En;q(m+ 1;λ)− En;q(λ)

2
, (4.22)

holds true in terms of the q-Apostol – Euler polynomials.

Proof. It is easy to observe that

(−λ)m+1
∞∑
j=0

(−λ)jqm+j+1e[m+j+1]qt +
∞∑
j=0

(−λ)jqje[j]qt =
m∑
j=0

(−1)j+1λjqje[j]qt. (4.23)

By (1.8), (1.9) and (4.23), via simple computation, we arrive at the desire (4.22) immediately.
Theorem 4.4 is proved.
Clearly, the above formula (4.22) is a q-extension of the formula in [20, p. 388] (Eq. (55)).
Setting λ = 1 in (4.22), then we have

Zn;q(m;n+ 1) =
m∑
j=0

(−1)j+1qj [j]nq = −(−1)mEn;q(m+ 1) + En;q
2

, (4.24)

which is a q-extension of the well-known formula in [20, p. 388] (Eq. (56)) and [1, p. 804], (23.1.4).

5. Some relationships between the q-Apostol – Euler polynomials and q-Hurwitz – Lerch
zeta-function. The Hurwitz – Lerch zeta-function Φ(z, s, a) defined by (cf., e.g., [29, p. 121]).

Φ(z, s, a) :=

∞∑
n=0

zn

(n+ a)s

(
a ∈ C \ Z−0 ; s ∈ C when |z| < 1; R(s) > 1 when |z| = 1

)
contains, as its special cases, not only the Riemann and Hurwitz (or generalized) zeta-functions

ζ(s) := Φ(1, s, 1) = ζ (s, 1) =
1

2s − 1
ζ

(
s,

1

2

)
,

ζ(s, a) := Φ(1, s, a) =

∞∑
n=0

1

(n+ a)s
, R (s) > 1, a /∈ Z−0 ,

and the Lerch zeta-function:

ls(ξ) :=

∞∑
n=1

e2nπiξ

ns
= e2πiξ Φ

(
e2πiξ, s, 1

)
, ξ ∈ R, R(s) > 1,

but also such other functions as the polylogarithmic function:

Lis(z) :=
∞∑
n=1

zn

ns
= z Φ(z, s, 1)

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1114 Q.-M. LUO(
s ∈ C when |z| < 1; R(s) > 1 when |z| = 1

)
and the Lipschitz – Lerch zeta-function (cf. [29, p. 122], Eq. 2.5 (11)):

φ(ξ, a, s) :=
∞∑
n=0

e2nπiξ

(n+ a)s
= Φ

(
e2πiξ, s, a

)
=: L (ξ, s, a)

(
a ∈ C \ Z−0 ; R(s) > 0 when ξ ∈ R \Z; R(s) > 1 when ξ ∈ Z

)
.

We define the q-Hurwitz – Lerch zeta-functions as follows:
Definition 5.1. For R(a) > 0, q-Hurwitz – Lerch zeta-function is defined by

Φq(z, s, a) :=
∞∑
n=0

znqn+a[
n+ a

]s
q

, R (a) > 0, a /∈ Z−0 .

Theorem 5.1. The following relationship:

En;q (a;λ) = 2Φq(−λ,−n, a), n ∈ N, |λ| ≤ 1, a ∈ C \ Z−0 , (5.1)

holds true between the q-Apostol – Euler polynomials and the q-Hurwitz – Lerch zeta-function.

Proof. We differentiate the both sides of (1.9) with respect to the variable t yields that

En;q(a;λ) =
dn

dtn
Va;λ;q(t)

∣∣∣∣
t=0

= 2
∞∑
k=0

(−λ)kqk+a
dn

dtn

{
e[k+a]qt

} ∣∣∣∣
t=0

=

= 2
∞∑
k=0

(−λ)kqk+a
(
[k + a]q

)n
= 2

∞∑
k=0

(−λ)kqk+a[
k + a

]−n
q

.

Theorem 5.1 is proved.
Letting q → 1 in (5.1), we have the following corollary.
Corollary 5.1. The following relationship:

En (a;λ) = 2Φ(−λ,−n, a), n ∈ N, |λ| ≤ 1, a ∈ C \ Z−0 ,

holds true between the Apostol – Euler polynomials and Hurwitz – Lerch zeta-function.

On the other hand, we define an analogue of the Hurwitz zeta-function as follows:
Definition 5.2. For R(s) > 1, a /∈ Z−0 , L-function is defined by

L(s, a) :=

∞∑
n=0

(−1)n

(n+ a)s
.

Clearly, L(s, a) = φ

(
1

2
, a, s

)
= Φ(eπi, s, a) = L

(
1

2
, s, a

)
.

Next we define an q-extension of the L-function.
Definition 5.3. For R(s) > 1, a /∈ Z−0 , the q-L-function is defined by

Lq(s, a) :=

∞∑
n=0

(−1)nqn+a

[n+ a]sq
.

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q-APOSTOL – EULER POLYNOMIALS AND q-ALTERNATING SUMS 1115

In the same way, we can obtain the following relationships.
Theorem 5.2. The following relationship:

En,q (a) = 2Lq(−n, a), n ∈ N, a ∈ C \ Z−0 ,

holds true between the q-Euler polynomials and q-L-function.

Corollary 5.2. The following relationship:

En (a) = 2L(−n, a), n ∈ N, a ∈ C \ Z−0 ,

holds true between the Euler polynomials and the L-function.

We define an analogue of the Riemann zeta-function:
Definition 5.4. The l-function is defined by

l(s) :=

∞∑
n=1

(−1)n

ns
, R (s) > 1.

Obviously, l(s) = ls

(
1

2

)
= Lis(−1).

It follows that we define an q-extension of the l-function:
Definition 5.5. The q-l-function is defined by

lq(s) :=

∞∑
n=1

(−1)nqn

[n]sq
, R (s) > 1.

Similarly, we can obtain the following explicit relationship:
Theorem 5.3. The following relationship:

En,q = 2lq(−n), n ∈ N,

holds true between the q-Euler numbers and q-l-function.

Corollary 5.3. The following relationship:

En = 2 l(−n), n ∈ N,

holds true between the Euler numbers and l-function.

6. Some explicit relationships between the q-Apostol – Bernoulli and q-Apostol – Euler poly-
nomials. In this section, we will investigate some relationships between the q-Apostol – Bernoulli
and q-Apostol – Euler polynomials. We also obtain an q-extension of Howard’s result.

It is easy to observe that

2

∞∑
n=0

λ2nq2n+xe[2n+x]qt −
∞∑
n=0

λnqn+xe[n+x]qt =
∞∑
n=0

(−λ)nqn+xe[n+x]qt.

By (1.7) and (1.9), via the simple computation, we obtain

En;q(x;λ) =
2

n+ 1

[
Bn+1;q(x;λ)− 2[2]nqBn+1;q2

(x
2

;λ2
)]
, (6.1)

which is just an q-extension of the formula of Luo and Srivastava (see [19, p. 636], Eq. (38))

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1116 Q.-M. LUO

En(x;λ) =
2

n+ 1

[
Bn+1(x;λ)− 2n+1Bn+1

(x
2

;λ2
)]
.

Setting x = 0 in (6.1), we get

En;q(λ) =
2

n+ 1

[
Bn+1;q(λ)− 2[2]nqBn+1;q2

(
λ2
)]
. (6.2)

Putting λ = 1 in (6.1), we have

En;q(x) =
2

n+ 1

[
Bn+1;q(x)− 2[2]nqBn+1;q2

(x
2

)]
, (6.3)

which is an q-extension of the well-known formula (see [1])

En(x) =
2

n+ 1

[
Bn+1(x)− 2n+1Bn+1

(x
2

)]
.

Remark 6.1. If taking x = 0 in (6.3), we obtain

En;q =
2

n+ 1

(
Bn+1;q − 2[2]nqBn+1;q2

)
is an q-extension of the formula (see [1, p. 805] (Entry (23.1.20)) and [25, p. 29])

En =
2

n+ 1
(1− 2n+1)Bn+1.

Remark 6.2. By (4.3) and (6.2), we easily obtain the following explicit recursive formula for
the q-Apostol – Bernoulli numbers:

[m]q
(
Bn;q(λ)− 2[2]n−1q Bn;q2(λ2)

)
=

n∑
k=0

(
n

k

)
[m]kqBk;qm(λm)Zn−k;q(m−1;n;λ)−[m]nqBn;qm(λm).

(6.4)
Remark 6.3. Setting λ = 1 in (6.4), we have

[m]q
(
Bn;q − 2[2]n−1q Bn;q2

)
=

n∑
k=0

(
n

k

)
[m]kqBk;qmZn−k;q(m− 1;n)− [m]nqBn;qm (m is even)

is an q-extension of Howard’s formula [13, p. 167] (Eq. (33))

m(1− 2n)Bn =
n−1∑
k=0

(
n

k

)
Bkm

kZn−k(m− 1) (m is even).

Remark 6.4. Letting q → 1 in (6.4), we obtain a new formula for the Apostol – Bernoulli
numbers as follows:

m
(
Bn(λ)− 2nBn(λ2)

)
=

n∑
k=0

(
n

k

)
mkBk(λm)Zn−k(m− 1;λ)−mnBn(λm) (m is even).

(6.5)
Obviously, by setting λ = 1 in (6.5) we get a new recurrence formula for the Bernoulli numbers:

Bn =
n∑
k=0

(
n

k

)
mk−1Zn−k(m− 1)

1− 2n +mn−1 Bk (m is even).

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q-APOSTOL – EULER POLYNOMIALS AND q-ALTERNATING SUMS 1117

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Received 13.05.11,
after revision — 20.05.13

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