Volume 2008, Article ID 270713,19pages 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-adicq-L-function of two variables, which is recently constructed by Kim 2005and 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 Fox2000, Gunaratne1995, and Young1999, 2001.
Copyrightq2008 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
Forn∈ Z,n ≥ 0, Bernoulli numbersBn originally arise in the study of finite sums of a given
power of consecutive integers. They are given byB0 1, B1 −1/2,B2 1/6, B3 0, B4 −1/30, . . ., withB2n10 for oddn >1, and
Bn−n1
1
n−1
m0
n1 m
Bm, 1.1
for alln≥1. In the symbolic notation, Bernoulli numbers are given recursively by
B1n−Bn δn,1, 1.2
with the usual convention about replacingBj byBj, where δn,1 is the Kronecker symbol. The Bernoulli polynomialsBnzcan be expressed in the form
Bnz Bzn n
m0
n m
for an indeterminatez. The generating functions of these numbers and polynomials are given, respectively, by
Ft t et−1
∞
n0 Bnt
n
n!,
Fz, t ett− 1e
zt∞ n0
Bnzt n
n!,
1.4
for|t|< 2π. One of the notable facts about Bernoulli numbers and polynomials is the relation between the Riemann and the Hurwitzor generalizedzeta functions.
Theorem 1.1see1,2. For every integern≥1,
ζ1−n −Bn
n , ζ1−n, z − Bnz
n , 1.5
whereζsandζs, zare the Riemann and the Hurwitz (or generalized) zeta functions, defined, re-spectively, by
ζs ∞
m1 1
ms, ζs, z
∞
m0 1
mzs, 1.6
withs∈C,s>1, andz∈Cwithz>0.
Among various generalizations of Bernoulli numbers and polynomials, generalization with a primitive Dirichlet characterχhas a special case of attention.
Definition 1.2see2,3. For a primitive Dirichlet characterχhaving conductorf ∈Z,f ≥1, the generalized Bernoulli numbersBn,χand polynomialsBn,χzassociated withχare defined
by
Fχt f
a1
χateat
eft−1
∞
n0 Bn,χt
n
n!,
Fχz, t f
a1
χateazt
eft−1
∞
n0
Bn,χzt n
n!,
1.7
respectively, for|t|<2π/f.
Whenχ1, the classical Bernoulli numbers and polynomials are obtained in thatBn,1
−1nBnandBn,1z −1nBn−z. The generalized Bernoulli numbers and polynomials can be expressed in terms of Bernoulli polynomials as
Bn,χfn−1 f
a1
χaBn
a
f
,
Bn,χz fn−1 f
a1
χaBn
az
f
.
Given a primitive Dirichlet characterχhaving conductorf,the DirichletL-function as-sociated withχis defined by1,2
Ls, χ ∞
m1 χm
ms , 1.9
wheres∈C, Res> 1. It is well known2thatLs, χmay be analytically continued to the whole complex plane, except for a simple pole ats 1 whenχ 1, in which case it reduces to Riemann zeta function,ζs Ls,1. The generalized Bernoulli numbers share a particular relationship with the DirichletL-function in that
L1−n, χ −Bn,χ
n , 1.10
forn∈Z,n≥1.
Let pbe a fixed prime number. Throughout this paper,Zp,Qp,C, and Cp will,
respec-tively, 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 ofQp. Let| · |pdenote thep-adic
ab-solute value on Qp, normalized so that |p|p p−1. Letp∗ 4 ifp 2 andp∗ p otherwise.
Note that there existφp∗distinct solutions, modulop∗, to the equationxφp∗−10, and each solution must be congruent to one of the valuesa ∈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 ≡ amodp∗Zp. Lettingwa 0 fora ∈ Zsuch that a, p/1, it can be seen thatwis
actually a Dirichlet character having conductorfw p∗, called the Teichm ¨uller character. Let
x wx x. Then x ≡ 1modp∗Zp. In the sense of product of characters, letχn χw−n.
This implies thatfχn | fp∗. Sinceχ χnwn,f | fχnp∗is also true. Thus,f andfχn differ by a factor that is a power ofp.
During the development ofp-adic analysis, researches were made to derive a meromor-phic function, defined over thep-adic number field, that would interpolate the same, or at least similar values as the DirichletL-function at nonpositive integers. In4, Kubota and Leopoldt proved the existence of such a function, considered as p-adic equivalent of the Dirichlet L-function.
Proposition 1.3 see3, 4. There exists a uniquep-adic meromorphic (analytic ifχ / 1) function Lps, χ,s∈Zp, for which
Lp1−n, χ 1−χnppn−1L1−n, χn, 1.11
forn∈Z,n≥1.
By1.10, this function yields the values
Lp1−n, χ −1n1−χnppn−1Bn,χn, 1.12
forn∈Z,n≥1. Since the time of the work of Kubota and Leopoldt, many mathematicians have derived the existence and generalizations of Lps, χ by various means 5–12. In particular,
Washington 11derived the function by elementary means and expressed it in an explicit form.
LetDdenote the region
D s∈Cp :|s−1|p <|p|1p/p−1|p∗|−1p
Theorem 1.4see11. LetFbe a positive integer multiple ofp∗andf, and let
Lps, χ s−11F1 F
a1
a,p1
χa a1−s∞ m0
1−s m
F a
m
Bm. 1.14
Then,Lps, χis analytic fors ∈ D, whenχ /1, and meromorphic fors ∈ D, with a simple pole at
s1, having residue1−1/p, whenχ1. Furthermore, for eachn∈Z,n≥1,
Lp1−n, χ −1n1−χnppn−1Bn,χn. 1.15
Thus,Lps, χvanishes identically ifχ−1 −1.
In6, Fox derived ap-adic functionLps, z, χ, wherez∈ Cp,|z|p ≤ 1, ands ∈D, that
interpolates the values
Lp1−n, z, χ −n1Bn,χn
p∗z−χ
nppn−1Bn,χn
p−1p∗z, 1.16
for positive integersn. By applying the method that Washington used to deriveTheorem 1.4, Fox7obtainedLps, z, χby elementary means and expressed it in an explicit form.
Theorem 1.5see7. LetFbe a positive integer multiple ofp∗andf, and let
Lps, z, χ s−1
1 χ−1
F
F
a1
a,p1
χaa−p∗z1−s
×∞
m0
1−s
m a−Fp∗z
m
Bm.
1.17
Then,Lps, z, χis analytic forz ∈ Cp,|z|p ≤ 1, provided thats ∈ D, except fors /1whenχ 1.
Also, ifz∈Cp,|z|p ≤ 1, this function is analytic fors ∈Dwhenχ /1, and meromorphic fors ∈D,
with a simple pole ats1, having residue1−1/p, whenχ1. Furthermore, for eachn∈Z,n≥1,
Lp1−n, z, χ −n1Bn,χn
p∗z−χ
nppn−1Bn,χn
p−1p∗z. 1.18
In12, Young gavep-adic integral representations for the two-variablep-adicL-function introduced by Fox. These representations leaded to generalizations of some formulas of Dia-mond13,14and of Ferrero and Greenberg15forp-adicL-functions in terms of thep-adic gamma and log gamma functions. But, his work was restricted to character χ such that the conductor ofχ1is not a power ofp. The explicit formula given inTheorem 1.5by Fox yielded to derive formulas similar to that obtained by Young, but for all primitive Dirichlet characterχ. In16, Carlitz definedq-extensions of Bernoulli numbers and polynomials, and proved properties generalizing those satisfied by Bn and Bnz. When talking about q-extensions, q
can be considered as an indeterminate, a complex numberq ∈Cor ap-adic numberq ∈Cp. If
thatqx explogpqfor|x|p ≤1, where log is the Iwasawap-adic logarithm functionsee3, Chapter 4.
Theq-Bernoulli numbersβn,q,n∈Z,n≥0, are usually defined by
β0,q qlog−1q,qβq1n−βn,q δn,1, 1.19
where the usual convention about replacingβjqbyβj,qin the binomial expansion is understood
8,17–24. It follows from1.19that
βn,q 1
1−qn
n
i0
n i
−1i i
iq, 1.20
where it is understood that fori0, the functioni/iq 1. We use the notation
xq 1−qx
1−q , 1.21
so that limq→1xq xfor anyx∈Cin the complex case andx∈Cpwith|x|p ≤1 in thep-adic
case. In8,9, Kim definedq-Bernoulli polynomialsβn,qz,n∈Z,n≥0, as
βn,qz qzβqzn
n
m0
n m
qmzβ
m,qznq−m
1 1−qn
n
i0
n i
−1iqiz i
iq.
1.22
Some basic properties ofq-Bernoulli polynomialsβn,qzsimilar to those of Bernoulli
polyno-mialsBnzcan be deduced from1.22 see also25. For instance, we have
βn,q−11−z −1nqn−1βn,qz, 1.23
βn,q1z−βn,qz nqzznq−1, 1.24
βn,qzτ n
m0
n m
qmzβm,qτznq−m. 1.25
Letχbe a Dirichlet character with conductorf. The generalizedq-Bernoulli polynomials associated withχ,βn,q,χz,n∈Z,n≥0, are defined by8,9
βn,q,χz fnq−1 f
a1
χaβn,qf
az
f
. 1.26
Forz0,βn,q,χ0 βn,q,χare the generalizedq-Bernoulli numbers,
βn,q,χ fnq−1 f
a1
χaβn,qf
a
f
From1.25,1.26, and1.27,
βn,q,χz n
m0
n m
qmzβ
m,q,χznq−m. 1.28
An important property that the polynomials βn,q,χz satisfy is the following, which can be
proved by using1.24and1.26: Proposition 1.6. Form∈Z,m≥1,
βn,q,χmfz−βn,q,χz n mf
a1
χaqazazn−1
q 1.29
for alln∈Z,n≥1.
Note that forχ1i.e.,f1,z0, andq →1,Proposition 1.6reduces to
m
a1
an−1 1 n
Bn,1m−Bn,1
, 1.30
which is the well-known property of Bernoulli numbers and polynomials.
LetKbe an extension ofQp contained inCp. An infinite seriesan, an ∈ K, converges
in K if and only if |an|p → 0, as n → ∞. Let Kx andKxbe, respectively, the algebras
of formal power series and of polynomials inx. Then,Ax anxn ∈ Kxconverges at
xη,η∈Cp, if and only if|anηn|p →0, asn→ ∞. The following is a uniqueness property for
power series found in3.
Lemma 1.7. Let Ax, Bx ∈ Kx such that each converges in a neighborhood of 0 in Cp. If
Aηn Bηnfor a sequence{ηn},ηn / 0, inCpsuch thatηn→0, thenAx Bx.
Any positive integerncan be uniquely expressed in the form
n k
m0
ampm, 1.31
wheream∈Z, 0≤am≤p−1, form0,1, . . . , kandak / 0. For suchn, let
spn k
m0
am 1.32
be the sum of thep-adic digits ofnwithsp0 0. For anyn∈Z, letvpnbe the highest power
ofpdividingn. The functionvp is additive and relatessp by means of
vpn! n−ps−pn
1 1.33
for alln≥0. Forn≥1,1.33implies that
We denote a particular subring ofCpas
oa∈Cp:|a|p<1. 1.35
If z ∈ Cp such that |z|p ≤ |p|mp, where m ∈ Q, thenz ∈ pmo, and this can be also written as
z≡0modpmo. Let the setRbe defined as
Ra∈Cp :|a|p < p−1/p−1
. 1.36
Obviously,R ⊂ o. Since|1−q|p < p−1/p−1forq ∈ Cp, we have 1−q ∈ R, which implies that
q≡1modR. Let a:q aqw−1a. For the context in the sequel, an extension of a:qis needed. Sincewcan be considered as a Dirichlet character of conductorp∗,wap∗z wa fora∈Zwitha, p 1. Thus, ap∗z:qcan be defined by
ap∗z:q
ap∗z
q
wa . 1.37
Ifz∈Cpsuch that|z|p≤1, then for anya∈Z,
ap∗zq aqqap∗zq≡aqmodR. 1.38
Thus, ap∗z:q ≡1modp∗R.
LetFbe a positive integer multiple off andp∗. In9, Kim definedp-adicq-L-function of two variablesLp,qs, z, χas follows:
Lp,qs, z, χ s−11F1 q
F
a1
a,p1
χaap∗z:q1−s
×∞
m0
1−s m
βm,qFqap∗zm
F
ap∗z
m
qap∗z.
1.39
The analytic properties ofLp,qs, z, χare given by the following theorem.
Theorem 1.8see9. LetF be a positive multiple off andp∗and letLp,qs, z, χbe as in 1.39.
Then,Lp,qs, z, χis analytic forz ∈ Cp,|z|p ≤ 1, provided thats ∈ D, except for s 1 ifχ /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 ats 1 with residue1/FqqF −1/log q1−1/pif χ 1. Furthermore, forn∈Z,n≥1,
Lp,q1−n, z, χ −1n
βn,q,χn
p∗z−χ
nppnq−1βn,qp,χnp−1p∗z. 1.40
Kim 9 also gave a p-adic integral representation for the function Lp,qs, z, χ and
derived a q-extension of the generalized Diamond-Ferrero-Greenberg formula for the two-variable p-adic L-function in terms of p-adic gamma and log-gamma functions. In 5, first author derivedLp,qs, z, χby 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 by1.39.
The main motivation of this paper is to derive general classes of congruences for gener-alizedq-Bernoulli polynomials by making use of the functionLp,qs, z, χ. These classes are
variables, which generalizes Proposition 1.6 and thus the well-known formula for Bernoulli numbers and polynomials1.30.
2. Properties ofLp,qs, z, χ
Recall thatLp,qs, z, χ,z∈Cp,|z|p ≤1, interpolates the values
Lp,q1−n, z, χ −n1bnz, q, χ, 2.1
forn∈Z,n≥1, where
bnz, q, χ βn,q,χn
p∗z−χ
nppnq−1βn,qp,χnp−1p∗z. 2.2
Lemma 2.1. For alln∈Z,n≥1,
bn−z, q−1, χχ−1qn−1bnz, q, χ. 2.3
Proof. We use the method in26,27for the proof. First, consider the caseχn1, which implies
χwn. Then
bn−z, q−1, χβn,q−1,1
−p∗z−pn−1
q−1βn,q−p,1
−p−1p∗z
βn,q−1
1−p∗z− 1
qp−1n−1p
n−1
q βn,q−p1−p−1p∗z. 2.4
From1.23, we have
bn−z, q−1, χ −1nqn−1βn,qp∗z−
pnq−1
qp−1n−1−1
nqpn−1β
n,qpp−1p∗z
−1nqn−1βn,qp∗z−pnq−1βn,qpp−1p∗z.
2.5
Using1.24, we obtain
bn−z, q−1, χ
−1nqn−1 βn,q1p∗z−nqp∗zp∗znq−1−pnq−1βn,qp1p−1p∗zpnq−1nqpp
−1p∗z
p−1p∗zn−1
qp
−1nqn−1βn,q1p∗z−pnq−1βn,qp1p−1p∗z
−1nqn−1βn,q,1
p∗z−pn−1
q βn,qp,1
p−1p∗z
−1nqn−1bnz, q, χ.
2.6
Now, suppose thatχn / 1. Then, from1.26, we obtain
bn−z, q−1, χβn,q−1,χn
−p∗z−χ
nppqn−1−1βn,q−p,χn
−p−1p∗z
fχn
n−1
q−1 fχn
a1
χnaβn,q−fχn
a−p∗z fχn
−χnppnq−−11
fχn
n−1
q−p
fχn
a1
χnaβn,q−pfχn
a−p−1p∗z
fχn
fχn
n−1
q−1 fχn
a1
χnfχn−a
βn,q−fχn
f
χn−a−p∗z fχn
−χnppnq−−11
fχn
n−1
q−p
fχn
a1
χnfχn−a
βn,q−pfχn
f
χn−a−p−1p∗z fχn
fχn
n−1
q−1 fχn
a1
χn−aβn,q−fχn
1− ap ∗z
fχn
−χnppnq−−11
fχn
n−1
q−p
fχn
a1
χn−aβn,q−pfχn
1−ap −1p∗z
fχn
.
2.7
Using1.23, we have
bn−z, q−1, χ −1nqfχnn−1fχn
n−1
q−1 χn−1 fχn
a1
χnaβn,qfχn
ap∗z
fχn
−χnppqn−1−1−1n
qfχnn−1fχ n
n−1
q−p χn−1
fχn
a1
χnaβn,qpfχn
ap−1p∗z
fχn
−1nqn−1χn−1βn,q,χn
p∗z−χ
nppqn−1−1nqn−1χn−1βn,qp,χnp−1p∗z −1nqn−1χn−1bnz, q, χ.
2.8
Note thatχn−1 −1nχ−1. Thus, the lemma holds forχn /1. Since the lemma holds for
χn1 andχn / 1, the proof must be complete.
Using this result, we can prove the following theorem.
Theorem 2.2. Letz∈Cp,|z|p ≤1, ands∈D, except fors /1ifχ1. Then
Lp,q−1s,−z, χ χ−1q−sLp,qs, z, χ. 2.9
Proof. Letz∈Cp,|z|p ≤1, andn∈Z,n≥1. Since
Lemma 2.1implies that
Lp,q−11−n,−z, χ
−n1bn−z, q−1, χ−n1χ−1qn−1bnz, q, χ χ−1qn−1Lp,q1−n, z, χ,
2.11
and2.9holds for alls1−n,n∈Z,n≥1. Since the negative integers have 0 as a limit point,
Lemma 1.7implies thatTheorem 2.2holds for allsin any neighborhood about 0 common to the domains of the functions on either side of2.9. It is obvious that the domains, in the variable s, of the functions on both sides of2.9containD, except fors / 1 ifχ1. This completes the proof.
It is well known that the generalized Bernoulli polynomials associated with a Dirichlet characterχare important in regard to sums of consecutive integers, all of which raised to the same power. Proposition 1.6represents aq-extension of this property. In this section, we will give an extension ofProposition 1.6with the use ofLp,qs, z, χ.
For the characterχ, let F0 l cmf, p∗. Then,fχn | F0 for each n ∈ Z. Also, letF be a positive multiple ofpp∗−1F0.
Theorem 2.3. Letz∈Cp,|z|p ≤1, ands∈D, except fors /1ifχ1. Then
Lp,qs, zF, χ−Lp,qs, z, χ − p∗F
a1
a,p1
χ1aqap
∗z
ap∗z:q−s. 2.12
Proof. Letz∈Cp,|z|p ≤1, and letn∈Z,n≥1. From2.1, we have
Lp,q1−n, zF, χ−Lp,q1−n, z, χ −n1bnzF, q, χ−bnz, q, χ. 2.13
Equation2.2then implies that bnzF, q, χ−bnz, q, χ
βn,q,χn
p∗zp∗F−β
n,q,χn
p∗z−χ
nppnq−1
βn,qp,χnp−1p∗zp−1p∗F−βn,qp,χnp−1p∗z.
2.14
ByProposition 1.6, we can write
bnzF, q, χ−bnz, q, χ
np
∗F
a1
χnaqap∗z
ap∗zn−1
q −χnppnq−1n p−1p∗F
a1
χnaqpap
−1p∗z
ap−1p∗zn−1
qp
np
∗F
a1
χnaqap∗z
ap∗zn−1
q −n
p∗F
a1 p|a
χnaqap∗z
ap∗zn−1
q
np
∗F
a1
a,p1
χnaqap∗z
ap∗zn−1
q .
Therefore,
Lp,q1−n, zF, χ−Lp,q1−n, z, χ − p∗F
a1
a,p1
χnaqap∗zap∗znq−1. 2.16
Sinceχn χ1w−n−1, we can write
χnaap∗zqn−1 χ1aw−n−1a
ap∗zn−1
q χ1a
ap∗z:qn−1. 2.17
Thus,
Lp,q1−n, zF, χ−Lp,q1−n, z, χ − p∗F
a1
a,p1
χ1aqap
∗z
ap∗z:qn−1. 2.18
This implies that2.12is true for alls 1−n,n∈Z,n≥1. Since negative integers have 0 as a limit point,Lemma 1.7implies thatTheorem 2.3is true for allsin any neighborhood about 0 common to the domains of functions on both sides of2.12.
The domains, with respect tos, of the functions on the left of2.12containD, except for s / 1 ifχ1. Consider the sum
p∗F
a1
a,p1
χ1aqap
∗z
ap∗z:q−s p
∗F
a1
a,p1
χ1aqap
∗z
ap∗z:q−1ap∗z:q1−s.
2.19
This sum consists of the functions of the formqap∗z ap∗z:q1−s,a∈Z,a, p 1. Thus, it is sufficient to show that each such function is analytic onD;
ap∗z:q1−s can be written as
ap∗z:q1−s e1−slogp ap∗z:q ∞
m0 1
m!1−sm
logpap∗z:qm. 2.20
Since a p∗z : q ≡ 1modp∗R for all a ∈ Z, a, p 1 and z ∈ Cp, |z|p ≤ 1, we have
logp ap∗z:q ≡0modp∗R, which implies that
logp
ap∗z:q
p <p∗p|p|
1/p−1
p . 2.21
Now, by1.34and the definition of the domainD,
qap∗z 1
m!1−sm
logpap∗z:qm
p <|p|
1/p−1
p |p|pm−1/p−1p∗−pm|p|m/pp −1p∗mp|p|m/pp −1
|p|3m/p−1
p −→0,
2.22 asm→ ∞. So, whenevers∈D, the power series converges. Thus, the functions on either side of2.12have domains which containD, except possibly fors / 1 ifχ1. This completes the proof.
Corollary 2.4. Fors∈D, except fors /1ifχ1. Then
Lp,qs, F, χ−Lp,qs, χ − p∗F
a1
a,p1
3. Congruences for generalizedq-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 numberscf.2:
p−1Δ
cBnn ∈Zp, 3.1
wherec ∈ Z,c ≥ 1, c ≡ 0modp−1, andn ∈ Zis positive, even, andn /≡0modp−1. Here,Δcis the forward difference operator which operates on a sequence{xn}by
Δcxnxnc−xn. 3.2
The powersΔkc ofΔc are defined byΔ0c identity andΔkc Δc◦Δkc−1for positive integersk,
so that
Δk cxn
k
m0
k m
−1k−mxnmc. 3.3
More generally, it can be shown that
p−kΔk
cBnn ∈Zp, 3.4
wherek ∈Z,k≥1, andcandnare as above, but withn > k.
Kummer congruences for generalized Bernoulli numbersBn,χwere first regarded by
Car-litz28.
For positivec∈Z,c≡0modp−1,n, k ∈Z,n > k ≥1, andχsuch thatf fχ / pm,
wherem∈Z,m≥0,
p−kΔk c
Bn,χ
n ∈Zpχ. 3.5
Here,Zpχdenotes the ring of polynomials inχ, whose coefficients are inZp.
Shiratani29applied the operatorΔkc to−1−χnppn−1Bn,χn/nfor similarcandχ, and showed that Carlitz’s congruence is still true without the restrictionn > k, requiring only that n≥1. He also established that the divisibility conditions onccan be removed, and proved
p∗−kΔk
c1−χnppn−1Bn,χn n ∈Zpχ. 3.6
As an extension of the Kummer congruence, Gunaratne30,31showed that the value
p−kΔk c
1−χnppn−1Bn,χn n, 3.7
modulopZp, is independent ofnand
p−kΔk
c1−χnppn−1Bn,χn n ≡p−kΔkc1−χnppn−1Bn ,χn
n
if p > 3, c, n, k ∈ Z are positive, χ ωh, where h ∈ Z, h /≡0modp−1, n, k ∈ Z, k ≡ kmodp−1. Furthermore, by means of the binomial coefficient operator
p−1Δc k
xn k!1
k−1
j0
p−1Δ
c−j
xn, 3.9
it has been shown that for similar characterχ,
p−1Δ
c
k
1−χnppn−1Bn,χnn ∈Zp, 3.10
and this value, modulopZp, is independent ofn.
Fox6derived congruences similar to those above for the generalized Bernoulli poly-nomials without restrictions on the characterχ.
We now consider howCorollary 2.4can be utilized to derive a collection of congruences related to generalizedq-Bernoulli polynomials. LetF0l cmf, p∗andFbe a positive integer multiple ofpp∗−1F0. We incorporate the polynomial structure
Bnz, q, χ −n1
βn,q,χn
p∗z−χ
nppnq−1βn,qp,χnp−1p∗z 3.11
and the set structure
R∗x∈Z
p:|x|p < p−1/p−1
3.12
to derive the Kummer congruences for generalizedq-Bernoulli polynomials. Throughout, we assume thatq∈Zpwith|1−q|p< p−1/p−1, so thatq≡1modR∗.
Theorem 3.1. Letn,c,kbe positive integers andz∈pp∗−1F0R∗. Then, the quantity
p∗−kΔk
cBnz, q, χ−p∗−kΔckBn0, q, χ∈R∗χ, 3.13
and, modulop∗R∗χ, is independent ofn.
Proof. SinceΔcis a linear operator,Corollary 2.4implies that
Δk
cLp,q1−n, F, χ−ΔkcLp,q1−n, χ − p∗F
a1
a,p1
χ1aqaΔkc a:qn−1. 3.14
Thus,
ΔkcBnF, q, χ−ΔkcBn0, q, χ − p∗F
a1
a,p1
χ1aqa a: 1−1Δkc a:qn. 3.15
Note that
Δk
c a:qn k
m0
k m
Now, a:q ≡1modp∗R∗, which implies that a:qc ≡1modp∗R∗, and thusΔkc a:qn≡ 0modp∗kR∗. Therefore,
Δk
cBnF, q, χ−ΔkcBn0, q, χ≡0modp∗kR∗χ, 3.17
and so
p∗−kΔk
cBnF, q, χ−p∗−kΔckBn0, q, χ∈R∗χ. 3.18
Also, since a:qc ≡1modp∗R∗,
ΔkcBnF, q, χ−ΔkcBn0, q, χ − p∗F
a1
a,p1
χ1aqa a:qn−1
a:qc−1
p∗
k
3.19
implies that the value ofp∗−kΔckBnF, q, χ−p∗−kΔkcBn0, q, χ, modulop∗R∗χ, is
indepen-dent ofn.
Let z ∈ pp∗−1F0R∗. Since the set of positive integers in pp∗−1F0Z is dense in pp∗−1F
0R∗, there exists a sequence{zj}inpp∗−1F0Zwithzj >0 for eachj, such thatzj →z. Now,Bnz, q, χis a polynomial, which implies thatBnzj, q, χ→Bnz, q, χ. Therefore,
lim
j→∞
Δk
cBnzj, q, χ−ΔkcBn0, q, χ ΔkcBnz, q, χ−ΔckBn0, q, χ. 3.20
The left side of this equality is 0 modulop∗kR∗χ, which implies that
Δk
cBnz, q, χ−ΔkcBn0, q, χ≡0modp∗kR∗χ, 3.21
and so
p∗−kΔk
cBnz, q, χ−p∗−kΔckBn0, q, χ∈R∗χ. 3.22
Furthermore, for a positive integern,
lim
j→∞
p∗−kΔk
cBnzj, q, χ−p∗−kΔkcBn0, q, χ−p∗−kΔkcBnzj, q, χ−p∗−kΔkcBn0, q, χ p∗−kΔk
Sincezj ∈ pp∗−1F0Zfor allj, the quantity on the left must be 0 modulop∗R∗χ. Therefore, the valuep∗−kΔkcBnz, q, χ−p∗−kΔkcBn0, q, χ, modulop∗R∗χ, is independent ofn.
Theorem 3.2. Letn,c,k,kbe positive integers withk ≡kmodp−1and letz∈pp∗−1F0R∗. Then
p∗−kΔk
cBnz, q, χ−p∗−kΔkcBn0, q, χ≡p∗−k
Δk
c Bnz, q, χ−p∗−k
Δk
cBn0, q, χmodpR∗χ.
3.24
Proof. Letkandkbe positive integers such thatk≡kmodp−1. Without loss of generality, assume thatk≥k. From3.19,
p∗−kΔk
cBnF, q, χ−p∗−kΔkcBn0, q, χ
−p∗−kΔk
cBnF, q, χ−p∗−k
Δk
cBn0, q, χ
−p
∗F
a1
a,p1
χ1aqa a:qn−1
a:qc−1
p∗
k
−
a
:qc−1 p∗
k
−p
∗F
a1
a,p1
χ1aqa a:qn−1
a
:qc−1 p∗
k
a:qc−1 p∗
k−k
−1
.
3.25
Ifasuch that
a:qc−1/≡ 0modpp∗R∗, 3.26
then, sincek−k≡0modp−1, we have
a
:qc−1 p∗
k−k
−1≡0modpR∗. 3.27
Thus,
p∗−kΔk
cBnF, q, χ−p∗−kΔkcBn0, q, χ
≡p∗−kΔk
cBnF, q, χ−p∗−k
Δk
cBn0, q, χmodpR∗χ.
3.28
Now, letz∈pp∗−1F0R∗. Then, there exists a sequence{zj}inpp∗−1F0Zwithzj >0 for
eachj, such thatzj →z. Consider
lim
j→∞
p∗−kΔk
cBnzj, q, χ−p∗−kΔkcBn0, q, χ−p∗−k
Δk
c Bnzj, q, χ−p∗−k
Δk
cBn0, q, χ
p∗−kΔk
cBnz, q, χ−p∗−kΔkcBn0, q, χ−p∗−k
Δk
c Bnz, q, χ−p∗−k
Δk
c Bn0, q, χ.
3.29
The binomial coefficient operatorTkassociated to an operatorT is defined by writing the binomial coefficients
X k
XX−1· · ·X−k1
k! , 3.30
fork≥0 as a polynomial inX, and replacingXbyT.
In the proof of next theorem, we need special numbers, namely, the Stirling numbers of the first kindsn, k, which are defined by means of the generating function
log1tk
k!
∞
n0
sn, kn!tn, 3.31
for k ∈ Z,k ≥ 0. Since there is no constant term in the expansion of log 1t,sn, k 0 for 0 ≤ n < k. Also,sn, n 1, for alln ≥ 0. The numberssn, kare integers and satisfy the following relation related to binomial coefficients:
x k
n!1n
k0
sn, kxk. 3.32
For further information for Stirling numbers, we refer to32.
Theorem 3.3. Letn,c,kbe positive integers andz∈pp∗−1F0R∗. Then, the quantity
p∗−1Δc k
Bnz, q, χ−
p∗−1Δc k
Bn0, q, χ∈R∗χ, 3.33
and, modulop∗R∗χ, is independent ofn.
Proof. Since the binomial coefficients operator is a linear operator,Corollary 2.4implies that
p∗−1Δc k
Lp,q1−n, F, χ−
p∗−1Δc k
Lp,q1−n, χ − p∗F
a1
a,p1
χ1aqa
p∗−1Δc k
a:qn−1.
3.34
Then,
p∗−1Δc k
BnF, q, χ−
p∗−1Δc k
Bn0, q, χ − p∗F
a1
a,p1
χ1aqa a:q−1
p∗−1Δc k
a:qn.
3.35
Utilizing3.32, we can write
p∗−1Δ
c
k
a:qn 1 k!
k
m0
sk, mp∗−mΔm
c a:qn
k!1 k
m0
sk, mp∗−m a:qn a:qc−1m
which follows from3.16. Thus,
p∗−1Δc k
BnF, q, χ−
p∗−1Δc k
Bn0, q, χ
−p
∗F
a1
a,p1
χ1aqa a:q−1 a:qn
p∗−1 a:qc−1
k
.
3.37
Sincep∗−1 acq−1∈R∗for eacha∈Zwitha, p 1, we see that
a:qn
p∗−1 a:qc−1
k
∈R∗. 3.38
This then implies that
p∗−1Δ
c
k
BnF, q, χ−
p∗−1Δ
c
k
Bn0, q, χ∈R∗χ. 3.39
Furthermore, since a:qc ≡1modp∗R∗, the value of this quantity, modulop∗R∗χ, is inde-pendent ofn.
Now, letz ∈pp∗−1F0R∗, and let{zj}be a sequence inpp∗−1F0Z, withzj >0 for each j, such thatzj →z. Then,
lim
j→∞
p∗−1Δ
c
k
Bnzj, q, χ−
p∗−1Δc k
Bn0, q, χ
p∗−1Δc k
Bnz, q, χ−
p∗−1Δc k
Bn0, q, χ
3.40
must be inR∗χ. Now, letn∈Z,n>0, and consider
lim
j→∞
p∗−1Δc k
Bnzj, q, χ−
p∗−1Δc k
Bn0, q, χ
−
p∗−1Δc k
Bnzj, q, χ−
p∗−1Δc k
Bn0, q, χ p∗ −1Δ c k
Bnz, q, χ−
p∗−1Δ
c
k
Bn0, q, χ
−
p∗−1Δ
c
k
Bnz, q, χ−
p∗−1Δ
c
k
Bn0, q, χ
.
3.41
The quantity on the left must be 0 modulop∗R∗χ, which implies that the value of
p∗−1Δc k
Bnz, q, χ−
p∗−1Δc k
Bn0, q, χ, 3.42
Acknowledgment
This work was supported by Akdeniz University Scientific Research Project Unit.
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