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Some results for the q-Bernoulli, q-Euler numbers and polynomials
Advances in Difference Equations 2011, 2011:68 doi:10.1186/1687-1847-2011-68
Daeyeoul Kim ()
Min-Soo Kim ()
ISSN 1687-1847
Article type Research
Submission date 2 September 2011
Acceptance date 23 December 2011
Publication date 23 December 2011
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Some results for the q-Bernoulli, q-Euler numbers and
polynomials
Daeyeoul Kim
1
and Min-Soo Kim
∗2
1
National Institute for Mathematical Sciences, Doryong-dong, Yuseong-gu
Daejeon 305-340, South Korea
2
Department of Mathematics, KAIST, 373-1 Guseong-dong, Yuseong-gu,

Let | · |
p
be the p-adic norm on Q with |p|
p
= p
−1
. For convenience, | · |
p
will also
be used to denote the extended valuation on C
p
.
The Bernoulli polynomials, denoted by B
n
(x), are defined as
(1.1) B
n
(x) =
n

k=0

n
k

B
k
x
n−k
, n ≥ 0,

= 0 for any odd
integer k not smaller than 3. It is well known that the Bernoulli numbers can also
1
2
be expressed as follows
(1.3) B
k
= lim
N→∞
1
p
N
p
N
−1

a=0
a
k
(see [15,16]). Notice that, from the definition B
k
∈ Q, and these integrals are
independent of the prime p which used to compute them. The examples of (1.3)
are:
(1.4)
lim
N→∞
1
p
N


a=0
a
2
= lim
N→∞
1
p
N
p
N
(p
N
− 1)(2p
N
− 1)
6
=
1
6
= B
2
.
Euler numbers E
k
, k ≥ 0 are integers given by (cf. [17–19])
(1.5) E
0
= 1, E
k

x −
1
2

k−i
,
which holds for all nonnegative integers k and all real x, and which was obtained by
Raabe [21] in 1851. Setting x = 1/2 and normalizing by 2
k
gives the Euler numbers
(1.7) E
k
= 2
k
E
k

1
2

,
where E
0
= 1, E
2
= −1, E
4
= 5, E
6
= −61, . . . . Therefore, E

.
In Section 3, we obtain the generating functions of the q-Bernoulli, q-Euler num-
bers, and polynomials. We shall provide some basic formulas for the q-Bernoulli
and q-Euler polynomials which will be used to prove the main results of this article.
3
In Section 4, we construct the q-analogue of the Riemann’s ζ-functions, the
Hurwitz ζ-functions, and the Dirichlet’s L-functions. We prove that the value of
their functions at non-positive integers can be represented by the q-Bernoulli, q-
Euler numbers, and polynomials.
2. q-Bernoulli, q-Euler numbers and polynomials related to the Bosonic
and the Fermionic p-adic integral on Z
p
In this section, we provide some basic formulas for p-adic q-Bernoulli, p-adic
q-Euler numbers and polynomials which will be used to prove the main results of
this article.
Let UD(Z
p
, C
p
) denote the space of all uniformly (or strictly) differentiable C
p
-
valued functions on Z
p
. The p-adic q-integral of a function f ∈ UD(Z
p
) on Z
p
is
defined by

[x]
q
= x
for x ∈ Z
p
, where q tends to 1 in the region 0 < |q − 1|
p
< 1 (cf. [22,5,12]). The
bosonic p-adic integral on Z
p
is considered as the limit q → 1, i.e.,
(2.3) I
1
(f) = lim
N→∞
1
p
N
p
N
−1

a=0
f(a) =

Z
p
f(z)dµ
1
(z).

z +
1
2

k
q
in (2.4), respectively,
we get the following formulas for the p-adic q-Bernoulli and p-adic q-Euler numbers,
respectively, if q ∈ C
p
with 0 < |q − 1|
p
< 1 as follows
(2.5) B
k
(q) =

Z
p
[z]
k
q
dµ
1
(z) = lim
N→∞
1
p
N
p

−1

a=0

a +
1
2

k
q
(−1)
a
.
Remark 2.1. The q-Bernoulli numbers (2.5) are first defined by Kamano [3]. In
(2.5) and (2.6), take q → 1. Form (2.2), it is easy to that (see [17, Theorem 2.5])
B
k
(q) → B
k
=

Z
p
z
k
dµ
1
(z), E
k
(q) → E

and |q
i
− 1|
p
≤ |q − 1|
p
< 1,
where i ∈ Z. We easily see that if |q − 1|
p
< 1, then q
x
= 1 for x = 0 if and only if
q is a root of unity of order p
N
and x ∈ p
N
Z
p
(see [16]).
By (2.3) and (2.7), we obtain
(2.8)
I
1
(q
iz
) =
1
q
i
− 1

i
− 1)
m
− 1

=
1
q
i
− 1
lim
N→∞
1
p
N
∞

m=1

p
N
m

(q
i
− 1)
m
=
1
q

−1
m − 1

(q
i
− 1)
m
=
1
q
i
− 1
∞

m=1
(−1)
m−1
(q
i
− 1)
m
m
=
i log q
q
i
− 1
since the series log(1 + x) =

∞

+ 1
.
From (2.5), (2.6), (2.8) and (2.9), we obtain the following explicit formulas of B
k
(q)
and E
k
(q):
(2.10) B
k
(q) =
log q
(1 − q)
k
k

i=0

k
i

(−1)
i
i
q
i
− 1
,
(2.11) E
k

bosonic and the fermionic p-adic integral on Z
p
:
(2.12) B
k
(x, q) =

Z
p
[x + z]
k
q
dµ
1
(z) and E
k
(x, q) =

Z
p
[x + z]
k
q
dµ
−1
(z),
5
where q ∈ C
p
with 0 < |q − 1|

(2.14)
E
k
(x, q) =
k

i=0

k
i

E
i
(q)
2
i

x −
1
2

k−i
q
q
i(x−
1
2
)
=


k
q
= ([x]
q
+ q
x
[y]
q
)
k
and
(2.15)
[x + z]
k
q
=

1
2

k
q

[2x − 1]
q
1
2
+ q
x−
1

k−i
q
q
(x−
1
2
)i

1
2

−i
q

z +
1
2

i
q
(cf. [4,5]). The above formulas can be found in [7] which are the q-analogues of the
corresponding classical formulas in [17, (1.2)] and [23], etc. Obviously, put x =
1
2
in (2.14). Then
(2.16) E
k
(q) = 2
k
E

iy
B
i
(x, q)[y]
k−i
q
(k ≥ 0),
E
k
(x + y, q) =
k

i=0

k
i

q
iy
E
i
(x, q)[y]
k−i
q
(k ≥ 0).
6
Proof. Applying the relationship [x + y −
1
2
]

=

q
y

q
x−
1
2
2
E(q) +

x −
1
2

q

+ [y]
q

k
=
k

i=0

k
i


E
i
(x, q)[y]
k−i
q
.
Similarly, the first identity follows. 
Remark 2.3. From (2.12), we obtain the not completely trivial identities
lim
q →1
B
k
(x + y, q) =
k

i=0

k
i

B
i
(x)y
k−i
= (B(x) + y)
k
,
lim
q →1
E

i=0

k
i

q
i
[n]
i
q
B
i
(x, q
n
) = [n]
k
q
B
k

x +
1
n
, q
n

,
k

i=0

k

i=0

k
i

n
i
B
i
(x) = n
k
B
k

x +
1
n

,
k

i=0

k
i

n
i

N→∞
1
np
N
np
N
−1

a=0
[nx + a]
k
q
=
1
n
lim
N→∞
1
p
N
n−1

i=0
p
N
−1

a=0
[nx + na + i]
k

q
n
n−1

i=0
B
k

x + i
n
, q
n

.
If we put x = 0 in (2.18) and use (2.13), we find easily that
(2.19)
B
k
(q) =
[n]
k
q
n
n−1

i=0
B
k

i

n
)
=
1
n
k

j=0
[n]
j
q

k
j

B
j
(q
n
)
n−1

i=0
q
ij
[i]
k−j
q
.
Obviously, Equation (2.19) is the q-analogue of

(nx, q) =

Z
p
[nx + z]
k
q
dµ
−1
(z)
= lim
N→∞
n−1

i=0
p
N
−1

a=0
[nx + na + i]
k
q
(−1)
na+i
= [n]
k
q
n−1


i
E
k

x + i
n
, q
n

if n odd.
8
From (2.18) and (2.21), we can obtain Proposition 2.5 below.
Proposition 2.5 (Multiplication formulas). Let n be any positive integer. Then
B
k
(x, q) =
[n]
k
q
n
n−1

i=0
B
k

x + i
n
, q
n

q
(t) =
∞

k=0
B
k
(q)
t
k
k!
= e
B(q)t
and G
q
(t) =
∞

k=0
E
k
(q)
t
k
k!
= e
E(q)t
,
where the symbol B
k

t
,
G
q
(t) = 2
∞

m=0
(−1)
m
e
2[m+
1
2
]
q
t
.
Proof. Combining (2.10) and (3.1), F
q
(t) may be written as
F
q
(t) =
∞

k=0
log q
(1 − q)
k

k

i=1

k
i

(−1)
i
i
q
i
− 1

.
Here, the term with i = 0 is understood to be 1/log q (the limiting value of the
summand in the limit i → 0). Specifically, by making use of the following well-
known binomial identity
k

k − 1
i − 1

= i

k
i

(k ≥ i ≥ 1).
9


=
∞

k=0
1
(1 − q)
k
t
k
k!
+ log q
∞

k=1
k
(1 − q)
k
t
k
k!
∞

m=0
q
m
k−1

i=0


k−1
= e
t
1−q
+
t log q
1 − q
∞

m=0
q
m
∞

k=0

1 − q
m
1 − q

k
t
k
k!
.
Next, by (2.11) and (3.1), we obtain the result
G
q
(t) =
∞

2
k
∞

m=0
(−1)
m

1 − q
m+
1
2
1 − q

k
t
k
k!
= 2
∞

m=0
(−1)
m
∞

k=0

m +
1

k=0
B
k
(x, q)
t
k
k!
=
∞

k=0
(q
x
B(q) + [x]
q
)
k
t
k
k!
,
(3.3) G
q
(t, x) =
∞

k=0
E
k
(x, q)

q
(t, x) = e
[x]
q
t
F
q
(q
x
t) = e
t
1−q
+
t log q
1 − q
∞

m=0
q
m+x
e
[m+x]
q
t
.
10
Proof. From (3.1) and (3.2), we note that
F
q
(t, x) =

F
q
(q
x
t).
The second identity leads at once to Lemma 3.1. Hence, the lemma follows. 
Lemma 3.4.
G
q
(t, x) = e
[
x−
1
2
]
q
t
G
q

q
x−
1
2
2
t

= 2
∞


q
(k ≥ 0).
Proof. By applying (3.2) and Lemma 3.3, we obtain
(3.4)
F
q
(t, x) =
∞

k=0
B
k
(x, q)
t
k
k!
= 1 +
∞

k=0

1
(1 − q)
k+1
+ (k + 1)
log q
1 − q
∞

m=0

(k ≥ 1).
Hence,
B
k
(x + 1, q) − B
k
(x, q) = k
q
x
log q
q − 1
[x]
k−1
q
(k ≥ 1).
Similarly we prove the second part by (3.3) and Lemma 3.4. This proof is complete.

From Lemma 2.2 and Corollary 3.5, we obtain for any integer k ≥ 0,
[x]
k
q
=
1
k + 1
q − 1
q
x
log q

k+1

i
E
i
(x, q) + E
k
(x, q)

11
which are the q-analogues of the following familiar expansions (see, e.g., [7, p. 9]):
x
k
=
1
k + 1
k

i=0

k + 1
i

B
i
(x) and x
k
=
1
2

k

, q
n

=
nq
n(x−1)+1
log q
q − 1
k + 1
[n]
k+1
q
(1 + q[nx − n]
q
)
k
,
E
k

x +
1
n
, q
n

+ E
k

x +

x +
1 − n
n

= k

x +
1 − n
n

k−1
(k ≥ 1, n ≥ 1),
(3.7) E
k

x +
1
n

+ E
k

x +
1 − n
n

= 2

x +
1 − n

p
q
q −1
if k = 1
0 if k > 1,
E
0
(q) = 1,

q
−
1
2
E(q)
2
+

−
1
2

q

k
+

q
1
2
E(q)

and E
k
(1, q) =

q
1
2
2
E(q) +

1
2

q

k
.
This proof is complete. 
12
Remark 3.8. (1). We note here that quite similar expressions to the first identity of
Proposition 3.7 are given by Kamano [3, Proposition 2.4], Rim et al. [8, Theorem
2.7] and Tsumura [10, (1)].
(2). Letting q → 1 in Proposition 3.7, the first identity is the corresponding
classical formulas in [8, (1.2)]:
B
0
= 1, (B + 1)
k
− B
k


t=0
=

1
1 − q

k
−
k log q
q − 1
∞

m=0
q
m
[m]
k−1
q
,
(4.2) E
k
(q) =

d
dt

k
G
q

1−q

s−1
+
log q
q − 1
∞

m=1
q
m
[m]
s
q
,
ζ
q ,E
(s) =
2
2
s
∞

m=0
(−1)
m

m +
1
2

(−1)
m
(m + 1)
s
= ζ
E
(s).
(In [26, p. 1070], our ζ
E
(s) is denote φ(s).)
The values of ζ
q
(s) and ζ
q ,E
(s) at non-positive integers are obtained by the
following proposition.
Proposition 4.2. For k ≥ 1, we have
ζ
q
(1 − k) = −
B
k
(q)
k
and ζ
q ,E
(1 − k) = E
k−1
(q).
Proof. It is clear by (4.1) and (4.2). 

s
q
,
ζ
q ,E
(s, x) = 2
∞

m=0
(−1)
m
[m + x]
s
q
.
Note that ζ
q
(s, x) is a meromorphic function on C with only one simple pole at
s = 1 and ζ
q,E
(s, x) is a analytic function on C.
The values of ζ
q
(s, x) and ζ
q,E
(s, x) at non-positive integers are obtained by the
following proposition.
Proposition 4.4. For k ≥ 1, we have
ζ
q

F
q
(t, x) =
1
d
d−1

i=0
F
q
d

[d]
q
t,
x + i
d

,
G
q
(t, x) =
d−1

i=0
(−1)
i
G
q
d

t
1−q
d
+
1
d
d−1

i=0
[d]
q
t log q
d
1 − q
d
∞

n=0
q
nd+x+i
e
[nd+x+i]
q
t
=
1
d
d−1

i=0

d
[d]
q
t

,
where we use [n+(x+i)/d]
q
d
[d]
q
= [nd+x+i]
q
. So we have the first form. Similarly
the second form follows by Lemma 3.4. 
From (3.2), (3.3), Propositions 4.4 and 4.5, we obtain the following:
Corollary 4.6. Let d and k be any positive integer. Then
ζ
q
(1 − k, x) =
[d]
k
q
d
d−1

i=0
ζ
q
d

(x, t) and G
q,χ
(x, t) of the generalized q-Bernoulli and q-
Euler polynomials as follows:
(4.4)
F
q ,χ
(t, x) =
∞

k=0
B
k,χ
(x, q)
t
k
k!
=
1
f
f

a=1
χ(a)F
q
f

[f]
q
t,

a + x
f

if f odd,
where B
k,χ
(x, q) and E
k,χ
(x, q) are the generalized q-Bernoulli and q-Euler poly-
nomials, respectively. Clearly (4.4) and (4.5) are equal to
(4.6)
F
q ,χ
(t, x) =
t log q
1 − q
∞

m=0
χ(m)q
m+x
e
[m+x]
q
t
,
(4.7)
G
q,χ
(t, x) = 2


a=1
χ(a)te
(a+x)t
e
ft
− 1
=
∞

k=0
B
k,χ
(x)
t
k
k!
,
(4.9) G
χ
(t, x) = 2
f

a=1
(−1)
a
χ(a)e
(a+x)t
e
ft

f

,
(4.11)
E
k,χ
(x, q) = [f]
k
q
f

a=1
(−1)
a
χ(a)E
k

a + x
f
, q
f

if f odd.
By using the definitions of ζ
q
(s, x) and ζ
q ,E
(s, x), we can define the q-analogues
of Dirichlet’s L-function.
15

q
(s, x, χ) at non-positive integers.
Theorem 4.8. For k ≥ 1, we have
L
q
(1 − k, x, χ) = −
B
k,χ
(x, q)
k
and 
q
(1 − k, x, χ) = E
k−1,χ
(x, q).
Proof. Using Lemma 3.3 and (4.4), we obtain
∞

k=0
B
k,χ
(x, q)
t
k
k!
=
1
f
f


]
q
f
[f]
q
t

=
t log q
1 − q
∞

m=0
χ(m)q
m+x
e
[m+x]
q
t
,
where we use [n + (a + x)/f]
q
f
[f]
q
= [nf + a + x]
q
and

f

∞

m=0
χ(m)q
m+x
[m + x]
k−1
q
.
Hence for k ≥ 1
−
B
k,χ
(x, q)
k
=
log q
q − 1
∞

m=0
χ(m)q
m+x
[m + x]
k−1
q
= L
q
(1 − k, x, χ).
Similarly the second identity follows. This completes the proof. 

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