Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2008, Article ID 270713, 19 pages
doi:10.1155/2008/270713
Research Article
Congruences for Generalized
q-Bernoulli Polynomials
Mehmet Cenkci and Veli Kurt
Department of Mathematics, Akdeniz University, 07058 Antalya, Turkey
Correspondence should be addressed to Mehmet Cenkci, [email protected]
Received 9 December 2007; Accepted 15 February 2008
Recommended by Andrea Laforgia
In this paper, we give some further properties of p-adic q-L-function of two variables, which is
recently constructed by Kim 2005 and Cenkci 2006. One of the applications of these proper-
ties yields general classes of congruences for generalized q-Bernoulli polynomials, which are q-
extensions of the classes for generalized Bernoulli numbers and polynomials given by Fox 2000,
Gunaratne 1995,andYoung1999, 2001.
Copyright q 2008 M. Cenkci and V. Kurt. 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.
1. Introduction and primary concepts
For n ∈
Z, n ≥ 0, Bernoulli numbers B
n
originally arise in the study of finite sums of a given
power of consecutive integers. They are given by B
0
1, B
1
−1/2, B
2
, 1.2
with the usual convention about replacing B
j
by B
j
,whereδ
n,1
is the Kronecker symbol. The
Bernoulli polynomials B
n
z can be expressed in the form
B
n
zB z
n
n
m0
n
m
B
m
z
n−m
, 1.3
2 Journal of Inequalities and Applications
for an indeterminate z. The generating functions of these numbers and polynomials are given,
t
n
n!
,
1.4
for |t| < 2 π. One of the notable facts about Bernoulli numbers and polynomials is the relation
between the Riemann and the Hurwitz or generalized zeta functions.
Theorem 1.1 see 1, 2. For every integer n ≥ 1,
ζ1 − n−
B
n
n
,ζ1 − n, z−
B
n
z
n
, 1.5
where ζs and ζs, z are the Riemann and the Hurwitz (or generalized) zeta functions, defined, re-
spectively, by
ζs
∞
m1
1
m
s
,ζs, z
∞
n0
B
n,χ
t
n
n!
,
F
χ
z, t
f
a1
χate
azt
e
ft
− 1
∞
n0
B
n,χ
z
t
n
n!
,
,
B
n,χ
zf
n−1
f
a1
χaB
n
a z
f
.
1.8
M. Cenkci and V. Kurt 3
Given a primitive Dirichlet character χ having conductor f, the Dirichlet L-function as-
sociated with χ is defined by 1, 2
Ls, χ
∞
m1
χm
m
s
, 1.9
where s ∈
C,Res > 1. It is well known 2 that Ls, χ may be analytically continued to the
whole complex plane, except for a simple pole at s 1whenχ 1, in which case it reduces
−1
.Letp
∗
4ifp 2andp
∗
p otherwise.
Note that there exist φp
∗
distinct solutions, modulo p
∗
, to the equation x
φp
∗
− 1 0, and each
solution must be congruent to one of the values a ∈
Z,where1≤ a ≤ p
∗
− 1, a, p1. Thus,
by Hensel’s lemma, given a ∈
Z with a, p1, there exists a unique wa ∈ Z
p
such that
wa ≡ amod p
∗
Z
p
. Letting wa0fora ∈ Z such that a, p
/
1, it can be seen that w is
is also true. Thus, f and f
χ
n
differ by a
factor that is a power of p.
During the development of p-adic analysis, researches were made to derive a meromor-
phic function, defined over the p-adic number field, that would interpolate the same, or at least
similar values as the Dirichlet L-function at nonpositive integers. In 4, Kubota and Leopoldt
proved the existence of such a function, considered as p-adic equivalent of the Dirichlet L-
function.
Proposition 1.3 see 3, 4. There exists a unique p-adic meromorphic (analytic if χ
/
1) function
L
p
s, χ, s ∈ Z
p
,forwhich
L
p
1 − n, χ
1 − χ
n
pp
n−1
L
1 − n, χ
s ∈
C
p
: |s − 1|
p
< |p|
1/p−1
p
|p
∗
|
−1
p
. 1.13
4 Journal of Inequalities and Applications
Theorem 1.4 see 11. Let F be a positive integer multiple of p
∗
and f, and let
L
p
s, χ
1
s − 1
1
F
F
a1
1 − χ
n
pp
n−1
B
n,χ
n
. 1.15
Thus, L
p
s, χ vanishes identically if χ−1−1.
In 6, Fox derived a p-adic function L
p
s, z, χ,wherez ∈ C
p
, |z|
p
≤ 1, and s ∈ D,that
interpolates the values
L
p
1 − n, z, χ−
1
n
B
n,χ
n
1
s − 1
χ−1
F
F
a1
a,p1
χa
a − p
∗
z
1−s
×
∞
m0
1 − s
m
F
a − p
∗
z
m
n,χ
n
p
∗
z
− χ
n
pp
n−1
B
n,χ
n
p
−1
p
∗
z
. 1.18
In 12, Young gave p-adic integral representations for the two-variable p-adic L-function
introduced by Fox. These representations leaded to generalizations of some formulas of Dia-
mond 13, 14 and of Ferrero and Greenberg 15 for p-adic L-functions in terms of the p-adic
gamma and log gamma functions. But, his work was restricted to character χ such that the
conductor of χ
1
is not a power of p. The explicit formula given in Theorem 1.5 by Fox yielded
to derive formulas similar to that obtained by Young, but for all primitive Dirichlet character χ.
β
0,q
q − 1
log q
,
qβ
q
1
n
− β
n,q
δ
n,1
, 1.19
where the usual convention about replacing β
j
q
by β
j,q
in the binomial expansion is understood
8, 17–24. It follows from 1.19 that
β
n,q
1
1 − q
n
p
≤ 1inthep-adic
case. In 8, 9, Kim defined q-Bernoulli polynomials β
n,q
z, n ∈ Z, n ≥ 0, as
β
n,q
z
q
z
β
q
z
n
n
m0
n
m
q
mz
β
m,q
z
n−m
1 − z−1
n
q
n−1
β
n,q
z,
1.23
β
n,q
1 z − β
n,q
znq
z
z
n−1
q
,
1.24
β
n,q
z τ
n
m0
n
m
q
n,q,χ
are the generalized q-Bernoulli numbers,
β
n,q,χ
f
n−1
q
f
a1
χaβ
n,q
f
a
f
. 1.27
6 Journal of Inequalities and Applications
From 1.25, 1.26,and1.27,
β
n,q,χ
z
n
m0
n
m
Note that for χ 1 i.e., f 1, z 0, and q → 1, Proposition 1.6 reduces to
m
a1
a
n−1
1
n
B
n,1
m − B
n,1
, 1.30
which is the well-known property of Bernoulli numbers and polynomials.
Let
K be an extension of Q
p
contained in C
p
. An infinite series
a
n
,a
n
∈ K,converges
in
for a sequence {η
n
}, η
n
/
0,in
C
p
such that η
n
→ 0,thenAxBx.
Any positive integer n can be uniquely expressed in the form
n
k
m0
a
m
p
m
, 1.31
where a
m
∈ Z,0≤ a
m
≤ p − 1, for m 0, 1, ,kand a
k
/
0. For such n,let
s
n − 1
p − 1
. 1.34
M. Cenkci and V. Kurt 7
We denote a particular subring of
C
p
as
o
a ∈
C
p
: |a|
p
< 1
. 1.35
If z ∈
C
p
such that |z|
p
≤|p|
m
p
,wherem ∈ Q,thenz ∈ p
m
o, and this can be also written as
z ≡ 0mod p
for a ∈
Z with a, p1. Thus, a p
∗
z : q can be defined by
a p
∗
z : q
a p
∗
z
q
wa
. 1.37
If z ∈
C
p
such that |z|
p
≤ 1, then for any a ∈ Z,
a p
∗
z
q
F
q
F
a1
a,p1
χa
a p
∗
z : q
1−s
×
∞
m0
1 − s
m
β
m,q
F
q
ap
∗
zm
F
p
, |z|
p
≤ 1, then this function is analytic for s ∈ D if χ
/
1 and meromorphic
for s ∈ D with a simple pole at s 1 with residue 1/F
q
q
F
− 1/ log q1 − 1/p if χ 1.
Furthermore, for n ∈
Z, n ≥ 1,
L
p,q
1 − n, z, χ−
1
n
β
n,q,χ
n
p
∗
z
− χ
n
pp
tained as an application of the difference formula see 2.12 for the p-adic q-L-function of two
8 Journal of Inequalities and Applications
variables, which generalizes Proposition 1.6 and thus the well-known formula for Bernoulli
numbers and polynomials 1.30.
2. Properties of L
p,q
s, z, χ
Recall that L
p,q
s, z, χ, z ∈ C
p
, |z|
p
≤ 1, interpolates the values
L
p,q
1 − n, z, χ−
1
n
b
n
z, q, χ, 2.1
for n ∈
Z, n ≥ 1, where
b
n
z, q, χβ
n,q,χ
n
χ−1q
n−1
b
n
z, q, χ. 2.3
Proof. We use the method in 26, 27 for the proof. First, consider the case χ
n
1, which implies
χ w
n
.Then
b
n
− z, q
−1
,χ
β
n,q
−1
,1
− p
∗
z
− p
n−1
q
β
n,q
−p
1 − p
−1
p
∗
z
.
2.4
From 1.23,wehave
b
n
− z, q
−1
,χ
−1
n
q
n−1
β
n,q
p
∗
q
n−1
β
n,q
p
∗
z
− p
n−1
q
β
n,q
p
p
−1
p
∗
z
.
2.5
Using 1.24,weobtain
b
n
β
n,q
p
1p
−1
p
∗
z
p
n−1
q
n
q
p
p
−1
p
∗
z
p
−1
p
∗
z
−1
n
q
n−1
β
n,q,1
p
∗
z
− p
n−1
q
β
n,q
p
,1
p
−1
p
∗
z
−1
n
q
z
− χ
n
pp
n−1
q
−1
β
n,q
−p
,χ
n
− p
−1
p
∗
z
f
χ
n
n−1
q
−1
f
n−1
q
−p
f
χ
n
a1
χ
n
aβ
n,q
−pf
χ
n
a − p
−1
p
∗
z
f
χ
n
f
χ
n
χ
n
− χ
n
pp
n−1
q
−1
f
χ
n
n−1
q
−p
f
χ
n
a1
χ
n
f
χ
n
− a
a1
χ
n
−aβ
n,q
−f
χ
n
1 −
a p
∗
z
f
χ
n
− χ
n
pp
n−1
q
−1
f
χ
n
n−1
− z, q
−1
,χ
−1
n
q
f
χ
n
n−1
f
χ
n
n−1
q
−1
χ
n
−1
f
χ
n
a1
f
χ
n
n−1
q
−p
χ
n
−1
f
χ
n
a1
χ
n
aβ
n,q
pf
χ
n
a p
−1
p
∗
z
f
p
,χ
n
p
−1
p
∗
z
−1
n
q
n−1
χ
n
−1b
n
z, q, χ.
2.8
Note that χ
n
−1−1
n
χ−1. Thus, the lemma holds for χ
n
/
1. Since the lemma holds for
χ
n
1
n
b
n
z, q, χ. 2.10
10 Journal of Inequalities and Applications
Lemma 2.1 implies that
L
p,q
−1
1 − n, −z, χ
−
1
n
b
n
− z, q
−1
,χ
−
1
n
χ−1q
n−1
b
n
z, q, χχ−1q
n−1
−1
F
0
.
Theorem 2.3. Let z ∈
C
p
, |z|
p
≤ 1,ands ∈ D, except for s
/
1 if χ 1.Then
L
p,q
s, z F, χ − L
p,q
s, z, χ−
p
∗
F
a1
a,p1
χ
1
aq
ap
∗
z
z F, q, χ − b
n
z, q, χ
β
n,q,χ
n
p
∗
zp
∗
F
− β
n,q,χ
n
p
∗
z
− χ
n
pp
n−1
q
2.14
By Proposition 1.6,wecanwrite
b
n
z F, q, χ − b
n
z, q, χ
n
p
∗
F
a1
χ
n
aq
ap
∗
z
a p
∗
z
n−1
q
− χ
n
pp
n−1
p
n
p
∗
F
a1
χ
n
aq
ap
∗
z
a p
∗
z
n−1
q
− n
p
∗
F
a1
p|a
χ
n
aq
.
2.15
M. Cenkci and V. Kurt 11
Therefore,
L
p,q
1 − n, z F, χ − L
p,q
1 − n, z, χ−
p
∗
F
a1
a,p1
χ
n
aq
ap
∗
z
a p
∗
z
n−1
q
. 2.16
Since χ
a
a p
∗
z : q
n−1
. 2.17
Thus,
L
p,q
1 − n, z F, χ − L
p,q
1 − n, z, χ−
p
∗
F
a1
a,p1
χ
1
aq
ap
∗
z
a p
∗
z : q
p
∗
F
a1
a,p1
χ
1
aq
ap
∗
z
a p
∗
z : q
−1
a p
∗
z : q
1−s
. 2.19
This sum consists of the functions of the form q
ap
∗
z
a p
log
p
a p
∗
z : q
m
. 2.20
Since a p
∗
z : q≡1mod p
∗
R for all a ∈ Z, a, p1andz ∈ C
p
, |z|
p
≤ 1, we have
log
p
a p
∗
z : q≡0mod p
∗
R, which implies that
log
p
1 − s
m
log
p
a p
∗
z : q
m
p
< |p|
1/p−1
p
|p|
m−1/p−1
p
p
∗
−m
p
|p|
s, χ−
p
∗
F
a1
a,p1
χ
1
aq
a
a : q
−s
. 2.23
12 Journal of Inequalities and Applications
3. Congruences for generalized q-Bernoulli polynomials
Congruences related to classical and generalized Bernoulli numbers have found an amount
of interest. One of the most celebrated examples is the Kummer congruences for classical
Bernoulli numbers cf. 2:
p
−1
Δ
c
B
n
n
∈
Z
p
, 3.1
c
◦ Δ
k−1
c
for positive integers k,
so that
Δ
k
c
x
n
k
m0
k
m
−1
k−m
x
nmc
. 3.3
More generally, it can be shown that
p
−k
Δ
k
c
∈
Z
p
χ. 3.5
Here,
Z
p
χ denotes the ring of polynomials in χ, whose coefficients are in Z
p
.
Shiratani 29 applied the operator Δ
k
c
to −1 − χ
n
pp
n−1
B
n,χ
n
/n for similar c and χ,and
showed that Carlitz’s congruence is still true without the restriction n>k, requiring only that
n ≥ 1. He also established that the divisibility conditions on c can be removed, and proved
p
∗
−k
Δ
k
n
, 3.7
modulo p
Z
p
, is independent of n and
p
−k
Δ
k
c
1 − χ
n
pp
n−1
B
n,χ
n
n
≡ p
−k
Δ
k
c
1 − χ
∈ Z, k ≡
k
mod p − 1. Furthermore, by means of the binomial coefficient operator
p
−1
Δ
c
k
x
n
1
k!
k−1
j0
p
−1
Δ
c
− j
x
We now consider how Corollary 2.4 can be utilized to derive a collection of congruences
related to generalized q-Bernoulli polynomials. Let F
0
lcmf, p
∗
and F be a positive integer
multiple of pp
∗
−1
F
0
. We incorporate the polynomial structure
B
n
z, q, χ−
1
n
β
n,q,χ
n
p
∗
z
− χ
n
pp
assume that q ∈
Z
p
with |1 − q|
p
<p
−1/p−1
,sothatq ≡ 1mod R
∗
.
Theorem 3.1. Let n, c, k be positive integers and z ∈ pp
∗
−1
F
0
R
∗
. Then, the quantity
p
∗
−k
Δ
k
c
B
n
z, q, χ −
p,q
1 − n, χ−
p
∗
F
a1
a,p1
χ
1
aq
a
Δ
k
c
a : q
n−1
. 3.14
Thus,
Δ
k
c
B
n
F, q, χ − Δ
k
c
B
n
0,q,χ−
m
−1
k−m
a : q
nmc
a : q
n
a : q
c
− 1
k
. 3.16
14 Journal of Inequalities and Applications
Now, a : q≡1mod p
∗
R
∗
, which implies that a : q
c
≡ 1mod p
∗
R
∗
,andthusΔ
k
c
a : q
, 3.17
and so
p
∗
−k
Δ
k
c
B
n
F, q, χ −
p
∗
−k
Δ
k
c
B
n
0,q,χ ∈ R
∗
χ. 3.18
Also, since a : q
c
≡ 1mod p
− 1
p
∗
k
3.19
implies that the value of p
∗
−k
Δ
k
c
B
n
F, q, χ − p
∗
−k
Δ
k
c
B
n
0,q,χ,modulop
∗
R
∗
χ, is indepen-
dent of n.
0
Z with z
j
> 0foreachj, such that z
j
→ z.
Now, B
n
z, q, χ is a polynomial, which implies that B
n
z
j
,q,χ → B
n
z, q, χ. Therefore,
lim
j→∞
Δ
k
c
B
n
z
j
,q,χ
− Δ
k
c
B
n
0,q,χ ≡ 0
mod
p
∗
k
R
∗
χ
, 3.21
and so
p
∗
−k
Δ
k
c
B
n
z, q, χ − p
∗
−k
−k
Δ
k
c
B
n
0,q,χ
−
p
∗
−k
Δ
k
c
B
n
z
j
,q,χ−
p
∗
−k
Δ
k
−
p
∗
−k
Δ
k
c
B
n
z, q, χ −
p
∗
−k
Δ
k
c
B
n
0,q,χ
.
3.23
M. Cenkci and V. Kurt 15
Since z
0,q,χ,modulop
∗
R
∗
χ, is independent of n.
Theorem 3.2. Let n, c, k, k
be positive integers with k ≡ k
mod p − 1 and let z ∈ pp
∗
−1
F
0
R
∗
.
Then
p
∗
−k
Δ
k
c
B
n
z, q, χ−
Δ
k
c
B
n
0,q,χmod pR
∗
χ.
3.24
Proof. Let k and k
be positive integers such that k ≡ k
mod p −1. Without loss of generality,
assume that k ≥ k
.From3.19,
p
∗
−k
Δ
k
c
B
n
F, q, χ −
−k
Δ
k
c
B
n
0,q,χ
−
p
∗
F
a1
a,p1
χ
1
aq
a
a : q
n−1
a : q
c
− 1
p
− 1
p
∗
k
a : q
c
− 1
p
∗
k−k
− 1
.
3.25
If a such that
a : q
c
− 1
/
≡ 0
mod pp
∗
R
B
n
F, q, χ −
p
∗
−k
Δ
k
c
B
n
0,q,χ
≡
p
∗
−k
Δ
k
c
B
n
F, q, χ −
p
−1
F
0
Z with z
j
> 0for
each j, such that z
j
→ z. Consider
lim
j→∞
p
∗
−k
Δ
k
c
B
n
z
j
,q,χ−
p
∗
−k
k
c
B
n
0,q,χ
p
∗
−k
Δ
k
c
B
n
z, q, χ−
p
∗
−k
Δ
k
c
B
n
0,q,χ
3.29
Since the left side of this equality must be 0 modulo pR
∗
χ, the proof follows.
16 Journal of Inequalities and Applications
The binomial coefficient operator
T
k
associated to an operator T is defined by writing
the binomial coefficients
X
k
XX − 1 ···X − k 1
k!
, 3.30
for k ≥ 0 as a polynomial in X, and replacing X by T.
In the proof of next theorem, we need special numbers, namely, the Stirling numbers of
the first kind sn, k, which are defined by means of the generating function
log 1 t
k
k!
∞
n0
∗
. Then, the quantity
p
∗
−1
Δ
c
k
B
n
z, q, χ −
p
∗
−1
Δ
c
k
B
n
0,q,χ ∈ R
∗
χ, 3.33
p,q
1−n, χ−
p
∗
F
a1
a,p1
χ
1
aq
a
p
∗
−1
Δ
c
k
a : q
n−1
.
3.34
Then,
p
1
aq
a
a : q
−1
p
∗
−1
Δ
c
k
a : q
n
.
3.35
Utilizing 3.32,wecanwrite
p
∗
−1
Δ
c
k
−m
a : q
n
a : q
c
− 1
m
3.36
M. Cenkci and V. Kurt 17
which follows from 3.16. Thus,
p
∗
−1
Δ
c
k
B
n
F, q, χ −
p
∗
a : q
c
− 1
k
.
3.37
Since p
∗
−1
a
c
q
− 1 ∈ R
∗
for each a ∈ Z with a, p1, we see that
a : q
n
p
∗
−1
a : q
c
− 1
B
n
0,q,χ ∈ R
∗
χ. 3.39
Furthermore, since a : q
c
≡ 1mod p
∗
R
∗
, the value of this quantity, modulo p
∗
R
∗
χ,isinde-
pendent of n.
Now, let z ∈ pp
∗
−1
F
0
R
∗
,andlet{z
j
} be a sequence in pp
∗
−1
Δ
c
k
B
n
0,q,χ
p
∗
−1
Δ
c
k
B
n
z, q, χ −
p
∗
−1
Δ
j
,q,χ −
p
∗
−1
Δ
c
k
B
n
0,q,χ
−
p
∗
−1
Δ
c
k
B
n
z
n
z, q, χ −
p
∗
−1
Δ
c
k
B
n
0,q,χ
−
p
∗
−1
Δ
c
k
B
n
z, q, χ −
k
B
n
z, q, χ −
p
∗
−1
Δ
c
k
B
n
0,q,χ, 3.42
modulo p
∗
R
∗
χ, is independent of n.
18 Journal of Inequalities and Applications
Acknowledgment
This work was supported by Akdeniz University Scientific Research Project Unit.
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