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, cenkci@akdeniz.edu.tr
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 properties yields general classes of congruences for generalized q-Bernoulli polynomials, which are qextensions of the classes for generalized Bernoulli numbers and polynomials given by Fox 2000,
Gunaratne 1995, and Young 1999, 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 Bn originally arise in the study of finite sums of a given
power of consecutive integers. They are given by B0 1, B1 −1/2, B2 1/6, B3 0, B4 −1/30, . . ., with B2n1 0 for odd n > 1, and
n−1
1 Bn −
n 1 m0
n1
Bm ,
m
1.1
for all n ≥ 1. In the symbolic notation, Bernoulli numbers are given recursively by
B 1n − Bn δn,1 ,
1.2
with the usual convention about replacing B j by Bj , where δn,1 is the Kronecker symbol. The
Bernoulli polynomials Bn z can be expressed in the form
n
Bn z B z Bm zn−m ,
m
m0
n
n
1.3
2
Journal of Inequalities and Applications
for an indeterminate z. The generating functions of these numbers and polynomials are given,
respectively, by
Ft et
∞
t
tn
Bn ,
− 1 n0 n!
∞
t
tn
Fz, t t
ezt Bn z ,
n!
e −1
n0
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 −
Bn
,
n
ζ1 − n, z −
Bn z
,
n
1.5
where ζs and ζs, z are the Riemann and the Hurwitz (or generalized) zeta functions, defined, respectively, by
ζs ∞
1
,
s
m
m1
ζs, z ∞
1
s,
m0 m z
1.6
with s ∈ C, s > 1, and z ∈ C with z > 0.
Among various generalizations of Bernoulli numbers and polynomials, generalization
with a primitive Dirichlet character χ has a special case of attention.
Definition 1.2 see 2, 3. For a primitive Dirichlet character χ having conductor f ∈ Z, f ≥ 1,
the generalized Bernoulli numbers Bn,χ and polynomials Bn,χ z associated with χ are defined
by
Fχ t f
χateat
a1
eft
−1
∞
Bn,χ
n0
tn
,
n!
f
∞
χateazt tn
Fχ z, t Bn,χ z ,
ft
n!
e −1
n0
a1
1.7
respectively, for |t| < 2 π/f.
When χ 1, the classical Bernoulli numbers and polynomials are obtained in that Bn,1 −1n Bn and Bn,1 z −1n Bn −z. The generalized Bernoulli numbers and polynomials can
be expressed in terms of Bernoulli polynomials as
Bn,χ f n−1
f
χaBn
a1
Bn,χ z f
n−1
f
a1
a
,
f
χaBn
az
.
f
1.8
M. Cenkci and V. Kurt
3
Given a primitive Dirichlet character χ having conductor f, the Dirichlet L-function associated with χ is defined by 1, 2
∞
χm
Ls, χ ,
1.9
ms
m1
where s ∈ C, Re s > 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 1 when χ 1, in which case it reduces
to Riemann zeta function, ζs Ls, 1. The generalized Bernoulli numbers share a particular
relationship with the Dirichlet L-function in that
L1 − n, χ −
Bn,χ
,
n
1.10
for n ∈ Z, n ≥ 1.
Let p be a fixed prime number. Throughout this paper, Zp , Qp , C, and Cp will, respectively, denote the ring of p-adic integers, the field of p-adic rational numbers, the complex
number field, and the completion of the algebraic closure of Qp . Let | · |p denote the p-adic absolute value on Qp , normalized so that |p|p p−1 . Let p∗ 4 if p 2 and p∗ p otherwise.
∗
Note that there exist φp∗ distinct solutions, modulo p∗ , to the equation xφp − 1 0, and each
solution must be congruent to one of the values a ∈ Z, where 1 ≤ a ≤ p∗ − 1, a, p 1. Thus,
by Hensel’s lemma, given a ∈ Z with a, p 1, there exists a unique wa ∈ Zp such that
wa ≡ amod p∗ Zp . Letting wa 0 for a ∈ Z such that a, p /
1, it can be seen that w is
actually a Dirichlet character having conductor fw p∗ , called the Teichmüller character. Let
x wxx
. Then x
≡ 1mod p∗ Zp . In the sense of product of characters, let χn χw−n .
This implies that fχn | fp∗ . Since χ χn wn , f | fχn p∗ is also true. Thus, f and fχn differ by a
factor that is a power of p.
During the development of p-adic analysis, researches were made to derive a meromorphic 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 Lfunction.
Proposition 1.3 see 3, 4. There exists a unique p-adic meromorphic (analytic if χ /
1) function
Lp s, χ, s ∈ Zp , for which
Lp 1 − n, χ 1 − χn ppn−1 L 1 − n, χn ,
1.11
for n ∈ Z, n ≥ 1.
By 1.10, this function yields the values
1
1.12
1 − χn ppn−1 Bn,χn ,
n
for n ∈ Z, n ≥ 1. Since the time of the work of Kubota and Leopoldt, many mathematicians have
derived the existence and generalizations of Lp s, χ by various means 5–12. In particular,
Washington 11 derived the function by elementary means and expressed it in an explicit
form.
Let D denote the region
∗ −1
D s ∈ Cp : |s − 1|p < |p|1/p−1
.
1.13
|p
|
p
p
Lp 1 − n, χ −
4
Journal of Inequalities and Applications
Theorem 1.4 see 11. Let F be a positive integer multiple of p∗ and f, and let
F
∞
1 1
χaa
1−s
Lp s, χ s − 1 F a1
m0
1−s
m
m
F
Bm .
a
1.14
a,p1
Then, Lp s, χ is analytic for s ∈ D, when χ /
1, and meromorphic for s ∈ D, with a simple pole at
s 1, having residue 1 − 1/p, when χ 1. Furthermore, for each n ∈ Z, n ≥ 1,
Lp 1 − n, χ −
1
1 − χn ppn−1 Bn,χn .
n
1.15
Thus, Lp s, χ vanishes identically if χ−1 −1.
In 6, Fox derived a p-adic function Lp s, z, χ, where z ∈ Cp , |z|p ≤ 1, and s ∈ D, that
interpolates the values
Lp 1 − n, z, χ −
1
Bn,χn p∗ z − χn ppn−1 Bn,χn p−1 p∗ z ,
n
1.16
for positive integers n. By applying the method that Washington used to derive Theorem 1.4,
Fox 7 obtained Lp s, z, χ by elementary means and expressed it in an explicit form.
Theorem 1.5 see 7. Let F be a positive integer multiple of p∗ and f, and let
Lp s, z, χ F
1−s
1 χ−1 χa a − p∗ z
s − 1 F a1
a,p1
m
∞
F
1−s
×
Bm .
m
a − p∗ z
m0
1.17
Then, Lp s, z, χ is analytic for z ∈ Cp , |z|p ≤ 1, provided that s ∈ D, except for s / 1 when χ 1.
Also, if z ∈ Cp , |z|p ≤ 1, this function is analytic for s ∈ D when χ /
1, and meromorphic for s ∈ D,
with a simple pole at s 1, having residue 1 − 1/p, when χ 1. Furthermore, for each n ∈ Z, n ≥ 1,
Lp 1 − n, z, χ −
1
Bn,χn p∗ z − χn ppn−1 Bn,χn p−1 p∗ z .
n
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 Diamond 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 χ1 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.
In 16, Carlitz defined q-extensions of Bernoulli numbers and polynomials, and proved
properties generalizing those satisfied by Bn and Bn z. When talking about q-extensions, q
can be considered as an indeterminate, a complex number q ∈ C or a p-adic number q ∈ Cp . If
q ∈ C, then it is assumed that |q| < 1 and if q ∈ Cp , then it is assumed that |1 − q|p < p−1/p−1 , so
M. Cenkci and V. Kurt
5
that qx exp logp q for |x|p ≤ 1, where log is the Iwasawa p-adic logarithm function see 3,
Chapter 4.
The q-Bernoulli numbers βn,q , n ∈ Z, n ≥ 0, are usually defined by
β0,q n
q−1 , qβq 1 − βn,q δn,1 ,
log q
1.19
j
where the usual convention about replacing βq by βj,q in the binomial expansion is understood
8, 17–24. It follows from 1.19 that
n
i
1
n
−1i
βn,q ,
1.20
iq
1 − qn i0 i
where it is understood that for i 0, the function i/iq 1. We use the notation
xq 1 − qx
,
1−q
1.21
so that limq→1 xq x for any x ∈ C in the complex case and x ∈ Cp with |x|p ≤ 1 in the p-adic
case. In 8, 9, Kim defined q-Bernoulli polynomials βn,q z, n ∈ Z, n ≥ 0, as
n
βn,q z qz βq z
n
n
qmz βm,q zn−m
q
m
m0
n
i
1
n
.
−1i qiz
iq
1 − qn i0 i
1.22
Some basic properties of q-Bernoulli polynomials βn,q z similar to those of Bernoulli polynomials Bn z can be deduced from 1.22 see also 25. For instance, we have
βn,q−1 1 − z −1n qn−1 βn,q z,
1.23
βn,q 1 z − βn,q z nqz zn−1
q ,
n
n
qmz βm,q τzn−m
.
βn,q z τ q
m
m0
1.24
1.25
Let χ be a Dirichlet character with conductor f. The generalized q-Bernoulli polynomials
associated with χ, βn,q,χ z, n ∈ Z, n ≥ 0, are defined by 8, 9
βn,q,χ z fn−1
q
f
a1
χaβn,qf
az
.
f
1.26
For z 0, βn,q,χ 0 βn,q,χ are the generalized q-Bernoulli numbers,
βn,q,χ fn−1
q
f
a1
χaβn,qf
a
.
f
1.27
6
Journal of Inequalities and Applications
From 1.25, 1.26, and 1.27,
n
βn,q,χ z .
qmz βm,q,χ zn−m
q
m
m0
n
1.28
An important property that the polynomials βn,q,χ z satisfy is the following, which can be
proved by using 1.24 and 1.26:
Proposition 1.6. For m ∈ Z, m ≥ 1,
mf
βn,q,χ mf z − βn,q,χ z n χaqaz a zn−1
q
1.29
a1
for all n ∈ Z, n ≥ 1.
Note that for χ 1 i.e., f 1, z 0, and q → 1, Proposition 1.6 reduces to
m
an−1 a1
1
Bn,1 m − Bn,1 ,
n
1.30
which is the well-known property of Bernoulli numbers and polynomials.
Let K be an extension of Qp contained in Cp . An infinite series an , an ∈ K, converges
in K if and only if |an |p → 0, as n → ∞. Let Kx and Kx be, respectively, the algebras
of formal power series and of polynomials in x. Then, Ax an xn ∈ Kx converges at
x η, η ∈ Cp , if and only if |an ηn |p → 0, as n → ∞. The following is a uniqueness property for
power series found in 3.
Lemma 1.7. Let Ax, Bx ∈ Kx such that each converges in a neighborhood of 0 in Cp . If
Aηn Bηn for a sequence {ηn }, ηn /
0, in Cp such that ηn → 0, then Ax Bx.
Any positive integer n can be uniquely expressed in the form
n
k
am pm ,
1.31
m0
where am ∈ Z, 0 ≤ am ≤ p − 1, for m 0, 1, . . . , k and ak /
0. For such n, let
sp n k
am
1.32
m0
be the sum of the p-adic digits of n with sp 0 0. For any n ∈ Z, let vp n be the highest power
of p dividing n. The function vp is additive and relates sp by means of
vp n! n − sp n
p−1
1.33
n−1
.
p−1
1.34
for all n ≥ 0. For n ≥ 1, 1.33 implies that
vp n! ≤
M. Cenkci and V. Kurt
7
We denote a particular subring of Cp as
o a ∈ Cp : |a|p < 1 .
1.35
m
If z ∈ Cp such that |z|p ≤ |p|m
p , where m ∈ Q, then z ∈ p o, and this can be also written as
m
z ≡ 0mod p o. Let the set R be defined as
R a ∈ Cp : |a|p < p−1/p−1 .
1.36
Obviously, R ⊂ o. Since |1 − q|p < p−1/p−1 for q ∈ Cp , we have 1 − q ∈ R, which implies that
q ≡ 1mod R. Let a : q
aq w−1 a. For the context in the sequel, an extension of a : q
is
needed. Since w can be considered as a Dirichlet character of conductor p∗ , wa p∗ z wa
for a ∈ Z with a, p 1. Thus, a p∗ z : q
can be defined by
a p∗ z q
∗
.
1.37
ap z:q wa
If z ∈ Cp such that |z|p ≤ 1, then for any a ∈ Z,
a p∗ z q aq qa p∗ z q ≡ aq mod R.
1.38
Thus, a p∗ z : q
≡ 1mod p∗ R.
Let F be a positive integer multiple of f and p∗ . In 9, Kim defined p-adic q-L-function
of two variables Lp,q s, z, χ as follows:
Lp,q s, z, χ F
1−s
1 1
χa a p∗ z : q
s − 1 Fq a1
a,p1
m
1−s
F
∗
×
.
βm,qF qap zm
a p∗ z qap∗ z
m
m0
∞
1.39
The analytic properties of Lp,q s, z, χ are given by the following theorem.
Theorem 1.8 see 9. Let F be a positive multiple of f and p∗ and let Lp,q s, z, χ be as in 1.39.
Then, Lp,q s, z, χ is analytic for z ∈ Cp , |z|p ≤ 1, provided that s ∈ D, except for s 1 if χ /
1.
Moreover, if z ∈ Cp , |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/Fq qF − 1/ log q1 − 1/p if χ 1.
Furthermore, for n ∈ Z, n ≥ 1,
−1 ∗ 1
p ,χ
Lp,q 1 − n, z, χ − βn,q,χn p∗ z − χn ppn−1
β
1.40
p pz .
n,q
n
q
n
Kim 9 also gave a p-adic integral representation for the function Lp,q s, z, χ and
derived a q-extension of the generalized Diamond-Ferrero-Greenberg formula for the twovariable p-adic L-function in terms of p-adic gamma and log-gamma functions. In 5, first
author derived Lp,q s, z, χ by using convergent power series, a method developed by Iwasawa
3. Resulting function from this derivation is in closed form but satisfies same properties of
the function defined by 1.39.
The main motivation of this paper is to derive general classes of congruences for generalized q-Bernoulli polynomials by making use of the function Lp,q s, z, χ. These classes are obtained 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 Lp,q s, z, χ
Recall that Lp,q s, z, χ, z ∈ Cp , |z|p ≤ 1, interpolates the values
1
Lp,q 1 − n, z, χ − bn z, q, χ,
n
2.1
−1 ∗ p
bn z, q, χ βn,q,χn p∗ z − χn ppn−1
q βn,q ,χn p p z .
2.2
for n ∈ Z, n ≥ 1, where
Lemma 2.1. For all n ∈ Z, n ≥ 1,
bn − z, q−1 , χ χ−1qn−1 bn z, q, χ.
2.3
Proof. We use the method in 26, 27 for the proof. First, consider the case χn 1, which implies
χ wn . Then
−1 ∗
−p
bn − z, q−1 , χ βn,q−1 ,1 − p∗ z − pn−1
q−1 βn,q ,1 − p p z
βn,q−1 1 − p∗ z − q
1
−1 ∗
−p 1 − p
pn−1
pz .
q βn,q
n−1
p−1
2.4
From 1.23, we have
n−1
pq
n−1
bn − z, q−1 , χ −1n qn−1 βn,q p∗ z − −1n qp
βn,qp p−1 p∗ z
n−1
qp−1
−1 ∗ p
−1n qn−1 βn,q p∗ z − pn−1
.
q βn,q p p z
2.5
Using 1.24, we obtain
bn − z, q−1 , χ
n−1
∗ n−1 p p−1 p∗ z −1 ∗ n−1
−1 ∗
p 1p
β
p
z
p
n
q
p
z
−1n qn−1 βn,q 1p∗ z −nqp z p∗ z q −pn−1
p
p
n,q
q
q
q
−1 ∗
p
−1n qn−1 βn,q 1 p∗ z − pn−1
q βn,q 1 p p z
−1 ∗ p
−1n qn−1 βn,q,1 p∗ z − pn−1
q βn,q ,1 p p z
−1n qn−1 bn z, q, χ.
2.6
Since χ wn and w−1 1, the lemma holds for χn 1.
M. Cenkci and V. Kurt
9
Now, suppose that χn /
1. Then, from 1.26, we obtain
−1 ∗
−p
bn − z, q−1 , χ βn,q−1 ,χn − p∗ z − χn ppn−1
q−1 βn,q ,χn − p p z
fχn
−
f
χn
n−1 q−1
a1
χn aβn,q−fχn
n−1
χn ppn−1
q−1 fχn q−p
fχ n
f
χn
n−1 q−1
a1
χn f χ n
f χn
a1
a − p∗ z
fχn
χn aβn,q−pfχn
− a βn,q−fχn
a − p−1 p∗ z
fχn
f χn − a − p ∗ z
fχn
f
χn
n−1 − χn ppn−1
χn fχn − a βn,q−pfχn
f
−1
χ
−p
n
q
q
a1
2.7
fχn − a − p−1 p∗ z
fχn
fχn
n−1 a p∗ z
fχn q−1
χn −aβn,q−fχn 1 −
fχn
a1
f χn
n−1 a p−1 p∗ z
−pfχn
− χn ppn−1
1
−
.
χ
−aβ
f
−1
χ
n
−p
n q
n,q
q
fχn
a1
Using 1.23, we have
f
χn
n−1 n−1
bn − z, q−1 , χ −1n qfχn
fχn q−1 χn −1 χn aβn,qfχn
a1
a p∗ z
fχn
a p−1 p∗ z
fχn
a1
n n−1
−1n qn−1 χn −1βn,q,χn p∗ z − χn ppn−1
χn −1βn,qp ,χn p−1 p∗ z
q −1 q
f
χn
n−1
n fχn n−1 − χn ppn−1
fχn q−p χn −1 χn aβn,qpf χn
q−1 −1 q
−1n qn−1 χn −1bn z, q, χ.
2.8
1. Since the lemma holds for
Note that χn −1 −1n χ−1. Thus, the lemma holds for χn /
χn 1 and χn / 1, the proof must be complete.
Using this result, we can prove the following theorem.
Theorem 2.2. Let z ∈ Cp , |z|p ≤ 1, and s ∈ D, except for s / 1 if χ 1. Then
Lp,q−1 s, −z, χ χ−1q−s Lp,q s, z, χ.
2.9
Proof. Let z ∈ Cp , |z|p ≤ 1, and n ∈ Z, n ≥ 1. Since
1
Lp,q 1 − n, z, χ − bn z, q, χ.
n
2.10
10
Journal of Inequalities and Applications
Lemma 2.1 implies that
Lp,q−1 1 − n, −z, χ
1 1
− bn − z, q−1 , χ − χ−1qn−1 bn z, q, χ χ−1qn−1 Lp,q 1 − n, z, χ,
n
n
2.11
and 2.9 holds for all s 1 − n, n ∈ Z, n ≥ 1. Since the negative integers have 0 as a limit point,
Lemma 1.7 implies that Theorem 2.2 holds for all s in any neighborhood about 0 common to the
domains of the functions on either side of 2.9. It is obvious that the domains, in the variable
s, of the functions on both sides of 2.9 contain D, except for s /
1 if χ 1. This completes the
proof.
It is well known that the generalized Bernoulli polynomials associated with a Dirichlet
character χ are important in regard to sums of consecutive integers, all of which raised to the
same power. Proposition 1.6 represents a q-extension of this property. In this section, we will
give an extension of Proposition 1.6 with the use of Lp,q s, z, χ.
For the character χ, let F0 l cmf, p∗ . Then, fχn | F0 for each n ∈ Z. Also, let F be a
positive multiple of pp∗ −1 F0 .
1 if χ 1. Then
Theorem 2.3. Let z ∈ Cp , |z|p ≤ 1, and s ∈ D, except for s /
∗
Lp,q s, z F, χ − Lp,q s, z, χ −
p F
−s
∗ χ1 aqap z a p∗ z : q .
2.12
a1
a,p1
Proof. Let z ∈ Cp , |z|p ≤ 1, and let n ∈ Z, n ≥ 1. From 2.1, we have
Lp,q 1 − n, z F, χ − Lp,q 1 − n, z, χ −
1
bn z F, q, χ − bn z, q, χ.
n
2.13
Equation 2.2 then implies that
bn z F, q, χ − bn z, q, χ
βn,q,χn p∗ zp∗ F − βn,q,χn p∗ z − χn ppn−1
βn,qp ,χn p−1 p∗ zp−1 p∗ F − βn,qp ,χn p−1 p∗ z .
q
2.14
By Proposition 1.6, we can write
bn z F, q, χ − bn z, q, χ
∗
−1 ∗
p F
p
p F
n−1
ap−1 p∗ z n−1
∗ n χn aqap z a p∗ z q − χn ppn−1
n
χn a qp
a p−1 p∗ z qp
q
a1
a1
∗
∗
a1
a1
p F
p F
n−1
n−1
∗ ∗ n χn aqap z a p∗ z q − n χn aqap z a p∗ z q
n
p∗ F
a1
p|a
n−1
∗ χn aqap z a p∗ z q .
a,p1
2.15
M. Cenkci and V. Kurt
11
Therefore,
∗
p F
Lp,q 1 − n, z F, χ − Lp,q 1 − n, z, χ −
a1
n−1
∗ χn aqap z a p∗ z q .
2.16
a,p1
Since χn χ1 w−n−1 , we can write
n−1
n−1
n−1
χn a a p∗ z q χ1 aw−n−1 a a p∗ z q χ1 a a p∗ z : q
.
2.17
Thus,
∗
p F
Lp,q 1 − n, z F, χ − Lp,q 1 − n, z, χ −
n−1
∗ χ1 aqap z a p∗ z : q
.
2.18
a1
a,p1
This implies that 2.12 is true for all s 1 − n, n ∈ Z, n ≥ 1. Since negative integers have 0 as
a limit point, Lemma 1.7 implies that Theorem 2.3 is true for all s in any neighborhood about 0
common to the domains of functions on both sides of 2.12.
The domains, with respect to s, of the functions on the left of 2.12 contain D, except for
s
1
if
χ 1. Consider the sum
/
∗
p F
a1
a,p1
χ1 aq
ap∗ z
∗
ap z:q
−s
∗
p F
−1 1−s
∗ χ1 aqap z a p∗ z : q
.
a p∗ z : q
2.19
a1
a,p1
∗
This sum consists of the functions of the form qap z a p∗ z : q
1−s , a ∈ Z, a, p 1. Thus, it is
sufficient to show that each such function is analytic on D;
a p∗ z : q
1−s can be written as
∞
1−s
m
∗
1
e1−slogp ap z:q
.
1 − sm logp a p∗ z : q
a p∗ z : q
m!
m0
2.20
Since a p∗ z : q
≡ 1mod p∗ R for all a ∈ Z, a, p 1 and z ∈ Cp , |z|p ≤ 1, we have
logp a p∗ z : q
≡ 0mod p∗ R, which implies that
log a p∗ z : q < p∗ |p|1/p−1 .
2.21
p
p
p
p
Now, by 1.34 and the definition of the domain D,
∗ −m m/p−1 ∗ m m/p−1
m 1/p−1
ap∗ z 1
p |p|
p |p|
|p|m−1/p−1
1 − sm logp a p∗ z : q
q
< |p|p
p
p
p
p
p
m!
p
|p|3m/p−1
−→ 0,
p
2.22
as m → ∞. So, whenever s ∈ D, the power series converges. Thus, the functions on either side
of 2.12 have domains which contain D, except possibly for s / 1 if χ 1. This completes the
proof.
Corollary 2.4. For s ∈ D, except for s /
1 if χ 1. Then
∗
Lp,q s, F, χ − Lp,q s, χ −
p F
a1
a,p1
χ1 aqa 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 Δc
Bn
∈ Zp ,
n
3.1
where c ∈ Z, c ≥ 1, c ≡ 0mod p − 1, and n ∈ Z is positive, even, and n /
≡ 0mod p − 1.
Here, Δc is the forward difference operator which operates on a sequence {xn } by
Δc xn xnc − xn .
3.2
for positive integers k,
The powers Δkc of Δc are defined by Δ0c identity and Δkc Δc ◦ Δk−1
c
so that
k
k
k
Δc xn 3.3
−1k−m xnmc .
m
m0
More generally, it can be shown that
p−k Δkc
Bn
∈ Zp ,
n
3.4
where k ∈ Z, k ≥ 1, and c and n are as above, but with n > k.
Kummer congruences for generalized Bernoulli numbers Bn,χ were first regarded by Carlitz 28.
For positive c ∈ Z, c ≡ 0mod p − 1, n, k ∈ Z, n > k ≥ 1, and χ such that f fχ /
pm ,
where m ∈ Z, m ≥ 0,
p−k Δkc
Bn,χ
∈ Zp χ.
n
3.5
Here, Zp χ denotes the ring of polynomials in χ, whose coefficients are in Zp .
Shiratani 29 applied the operator Δkc to −1 − χn ppn−1 Bn,χn /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
∗ −k k Bn,χn
p
Δc 1 − χn ppn−1
∈ Zp χ.
n
3.6
As an extension of the Kummer congruence, Gunaratne 30, 31 showed that the value
Bn,χn
p−k Δkc 1 − χn ppn−1
,
n
3.7
modulo pZp , is independent of n and
Bn,χn
Bn ,χn ≡ p−k Δkc 1 − χn ppn −1
p−k Δkc 1 − χn ppn−1
mod pZp ,
n
n
3.8
M. Cenkci and V. Kurt
13
if p > 3, c, n, k ∈ Z are positive, χ ωh , where h ∈ Z, h /
≡ 0mod p − 1, n , k ∈ Z, k ≡
k mod p − 1. Furthermore, by means of the binomial coefficient operator
k−1
−1
1 p−1 Δc
3.9
p Δc − j xn ,
xn k
k! j0
it has been shown that for similar character χ,
Bn,χn
p−1 Δc ∈ Zp ,
1 − χn ppn−1
k
n
3.10
and this value, modulo pZp , is independent of n.
Fox 6 derived congruences similar to those above for the generalized Bernoulli polynomials without restrictions on the character χ.
We now consider how Corollary 2.4 can be utilized to derive a collection of congruences
related to generalized q-Bernoulli polynomials. Let F0 l cmf, p∗ and F be a positive integer
multiple of pp∗ −1 F0 . We incorporate the polynomial structure
Bn z, q, χ −
−1 ∗ 1
p
βn,q,χn p∗ z − χn ppn−1
q βn,q ,χn p p z
n
3.11
R∗ x ∈ Zp : |x|p < p−1/p−1
3.12
and the set structure
to derive the Kummer congruences for generalized q-Bernoulli polynomials. Throughout, we
assume that q ∈ Zp with |1 − q|p < p−1/p−1 , so that q ≡ 1mod R∗ .
Theorem 3.1. Let n, c, k be positive integers and z ∈ pp∗ −1 F0 R∗ . Then, the quantity
−k
∗ −k k
Δc Bn z, q, χ − p∗ Δkc Bn 0, q, χ ∈ R∗ χ,
p
3.13
and, modulo p∗ R∗ χ, is independent of n.
Proof. Since Δc is a linear operator, Corollary 2.4 implies that
∗
Δkc Lp,q 1
− n, F, χ −
Δkc Lp,q 1
p F
χ1 aqa Δkc a : q
n−1 .
3.14
χ1 aqa a : 1
−1 Δkc a : q
n .
3.15
− n, χ −
a1
a,p1
Thus,
∗
Δkc Bn F, q, χ
−
Δkc Bn 0, q, χ
−
p F
a1
a,p1
Note that
Δkc a
k
k
: q
−1k−m a : q
nmc a : q
n a : q
c − 1 .
m
m0
n
k
3.16
14
Journal of Inequalities and Applications
Now, a : q
≡ 1mod p∗ R∗ , which implies that a : q
c ≡ 1mod p∗ R∗ , and thus Δkc a : q
n ≡
0mod p∗ k R∗ . Therefore,
k
Δkc Bn F, q, χ − Δkc Bn 0, q, χ ≡ 0 mod p∗ R∗ χ ,
3.17
∗ −k k
−k
p
Δc Bn F, q, χ − p∗ Δkc Bn 0, q, χ ∈ R∗ χ.
3.18
and so
Also, since a : q
c ≡ 1mod p∗ R∗ ,
∗
Δkc Bn F, q, χ
−
Δkc Bn 0, q, χ
−
p F
a1
χ1 aqa a : q
n−1
a : q
c − 1
p∗
k
3.19
a,p1
implies that the value of p∗ −k Δkc Bn F, q, χ − p∗ −k Δkc Bn 0, q, χ, modulo p∗ R∗ χ, is independent of n.
Let z ∈ pp∗ −1 F0 R∗ . Since the set of positive integers in pp∗ −1 F0 Z is dense in
∗ −1
pp F0 R∗ , there exists a sequence {zj } in pp∗ −1 F0 Z with zj > 0 for each j, such that zj → z.
Now, Bn z, q, χ is a polynomial, which implies that Bn zj , q, χ → Bn z, q, χ. Therefore,
lim Δkc Bn zj , q, χ − Δkc Bn 0, q, χ Δkc Bn z, q, χ − Δkc Bn 0, q, χ.
j→∞
3.20
The left side of this equality is 0 modulo p∗ k R∗ χ, which implies that
k
Δkc Bn z, q, χ − Δkc Bn 0, q, χ ≡ 0 mod p∗ R∗ χ ,
3.21
p∗ −k Δkc Bn z, q, χ − p∗ −k Δkc Bn 0, q, χ ∈ R∗ χ.
3.22
and so
Furthermore, for a positive integer n ,
−k
−k
−k
−k
lim p∗ Δkc Bn zj , q, χ− p∗ Δkc Bn 0, q, χ − p∗ Δkc Bn zj , q, χ− p∗ Δkc Bn 0, q, χ
j→∞
∗ −k k
−k
−k
−k
p
Δc Bn z, q, χ − p∗ Δkc Bn 0, q, χ − p∗ Δkc Bn z, q, χ − p∗ Δkc Bn 0, q, χ .
3.23
M. Cenkci and V. Kurt
15
Since zj ∈ pp∗ −1 F0 Z for all j, the quantity on the left must be 0 modulo p∗ R∗ χ. Therefore,
the value p∗ −k Δkc Bn z, q, χ − p∗ −k Δkc Bn 0, q, χ, modulo p∗ 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 F0 R∗ .
Then
−k
−k −k ∗−k k
Δc Bn z, q, χ− p∗ Δkc Bn 0, q, χ ≡ p∗ Δkc Bn z, q, χ− p∗ Δkc Bn 0, q, χmod pR∗ χ.
p
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 . From 3.19,
−k
−k k
−k
−k k
Δc Bn F, q, χ − p∗
Δc Bn 0, q, χ
p∗ Δkc Bn F, q, χ − p∗ Δkc Bn 0, q, χ − p∗
∗
p F
−
a
n−1
a
n−1
χ1 aq a : q
a1
a : q
c − 1
p∗
a,p1
∗
−
p F
χ1 aq a : q
a1
a : q
c − 1
p∗
k
−
k a : q
c − 1
p∗
a : q
c − 1
p∗
k k−k
−1 .
a,p1
3.25
If a such that
≡ 0 mod pp∗ R∗ ,
a : q
c − 1 /
3.26
then, since k − k ≡ 0mod p − 1, we have
a : q
c − 1
p∗
k−k
− 1 ≡ 0 mod pR∗ .
3.27
Thus,
−k
∗ −k k
Δc Bn F, q, χ − p∗ Δkc Bn 0, q, χ
p
−k k
−k k
Δc Bn F, q, χ − p∗
Δc Bn 0, q, χ mod pR∗ χ .
≡ p∗
3.28
Now, let z ∈ pp∗ −1 F0 R∗ . Then, there exists a sequence {zj } in pp∗ −1 F0 Z with zj > 0 for
each j, such that zj → z. Consider
−k
−k −k k
∗−k k
lim
Δc Bn zj , q, χ− p∗ Δkc Bn 0, q, χ − p∗ Δkc Bn zj , q, χ− p∗
Δc Bn 0, q, χ
p
∗ −k k
−k
−k k
−k k
p
Δc Bn z, q, χ− p∗ Δkc Bn 0, q, χ − p∗
Δc Bn z, q, χ− p∗
Δc Bn 0, q, χ .
3.29
j→∞
Since the left side of this equality must be 0 modulo pR∗ χ, the proof follows.
16
Journal of Inequalities and Applications
T The binomial coefficient operator k associated to an operator T is defined by writing
the binomial coefficients
XX − 1 · · · X − k 1
X
,
3.30
k
k!
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 tk tn
sn, k ,
k!
n!
n0
3.31
for k ∈ Z, k ≥ 0. Since there is no constant term in the expansion of log 1 t, sn, k 0
for 0 ≤ n < k. Also, sn, n 1, for all n ≥ 0. The numbers sn, k are integers and satisfy the
following relation related to binomial coefficients:
n
1
x
sn, kxk .
3.32
k
n! k0
For further information for Stirling numbers, we refer to 32.
Theorem 3.3. Let n, c, k be positive integers and z ∈ pp∗ −1 F0 R∗ . Then, the quantity
−1
−1
p∗ Δc
p∗ Δc
Bn z, q, χ −
Bn 0, q, χ ∈ R∗ χ,
k
k
3.33
and, modulo p∗ R∗ χ, is independent of n.
Proof. Since the binomial coefficients operator is a linear operator, Corollary 2.4 implies that
∗
−1
−1
−1
p
F
p∗ Δc
p∗ Δc
p∗ Δc
a
χ1 aq
Lp,q 1−n, F, χ−
Lp,q 1−n, χ −
a : q
n−1 .
k
k
k
a1
a,p1
3.34
Then,
p∗ F
−1
−1
−1
p∗ Δc
p∗ Δc
p∗ Δc
−1
a
Bn F, q, χ−
Bn 0, q, χ −
a : q
n .
χ1 aq a : q
k
k
k
a1
a,p1
3.35
Utilizing 3.32, we can write
k
−1
−m m
1
p∗ Δc
a : q
n sk, m p∗
Δc a : q
n
k! m0
k
k
−m
m
1
sk, m p∗
a : q
n a : q
c − 1
k! m0
3.36
M. Cenkci and V. Kurt
17
which follows from 3.16. Thus,
−1
−1
p∗ Δc
p∗ Δc
Bn F, q, χ −
Bn 0, q, χ
k
k
p∗ F
−1
a : q
c − 1
p∗
−1
n
a
.
−
χ1 aq a : q
a : q
k
a1
3.37
a,p1
Since p∗ −1 a
cq − 1 ∈ R∗ for each a ∈ Z with a, p 1, we see that
−1
a : q
c − 1
p∗
n
a : q
∈ R∗ .
k
This then implies that
−1
−1
p∗ Δc
p∗ Δc
Bn F, q, χ −
Bn 0, q, χ ∈ R∗ χ.
k
k
3.38
3.39
Furthermore, since a : q
c ≡ 1mod p∗ R∗ , the value of this quantity, modulo p∗ R∗ χ, is independent of n.
Now, let z ∈ pp∗ −1 F0 R∗ , and let {zj } be a sequence in pp∗ −1 F0 Z, with zj > 0 for each
j, such that zj → z. Then,
−1 −1
p∗ Δc
p∗ Δc
lim
Bn zj , q, χ −
Bn 0, q, χ
j→∞
k
k
3.40
−1 −1 p∗ Δc
p∗ Δc
Bn z, q, χ −
Bn 0, q, χ
k
k
must be in R∗ χ. Now, let n ∈ Z, n > 0, and consider
−1 ∗ −1 Δc
p∗ Δc
p
lim
Bn zj , q, χ −
Bn 0, q, χ
j→∞
k
k
−1 −1
p∗ Δc
p∗ Δc
−
Bn zj , q, χ −
Bn 0, q, χ
k
k
−1 ∗ −1 Δc
p∗ Δc
p
Bn z, q, χ −
Bn 0, q, χ
k
k
−1 −1 p∗ Δc
p∗ Δc
−
Bn z, q, χ −
Bn 0, q, χ .
k
k
The quantity on the left must be 0 modulo p∗ R∗ χ, which implies that the value of
−1 −1 p∗ Δc
p∗ Δc
Bn z, q, χ −
Bn 0, q, χ,
k
k
modulo p∗ R∗ χ, is independent of n.
3.41
3.42
18
Journal of Inequalities and Applications
Acknowledgment
This work was supported by Akdeniz University Scientific Research Project Unit.
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