J
OURNAL DE
T
HÉORIE DES
N
OMBRES DE
B
ORDEAUX
G
LENN
J. F
OX
Kummer congruences for expressions involving
generalized Bernoulli polynomials
Journal de Théorie des Nombres de Bordeaux, tome
14, n
o1 (2002),
p. 187-204
<http://www.numdam.org/item?id=JTNB_2002__14_1_187_0>
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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques
Kummer
congruences
for
expressions involving
generalized
Bernoulli
polynomials
par GLENN J. FOX
RÉSUMÉ. Nous illustrons le fait
qu’une expression particulière,
impliquant
lespolynômes
de Bernoulligénéralisés,
satisfait unsystème
de congruences si et seulement si uneexpression
sem-blable, impliquant
les nombres de Bernoulligénéralisés,
satisfait les mêmes relations de congruence. Parmi ces relations se trouventles congruences de Kummer ainsi que des
généralisations
fourniespar Gunaratne.
ABSTRACT. We illustrate how a
particular
expression,involving
the
generalized
Bernoullipolynomials,
satisfies systems ofcon-gruence relations if and
only
if a similarexpression,
involving
thegeneralized
Bernoullinumbers,
satisfies the same congruencere-lations. These congruence relations include the Kummer
congru-ences, and recent extensions of the Kummer congruences
provided
by
Gunaratne.1. Introduction
Let X
be aprimitive
Dirichlet characterhaving
conductorf.
Thegener-alized Bernoulli
polynomials
associated with x, are definedby
thegenerating
functionThe
corresponding generalized
Bernoulli numbers can then be definedby
= If we let
Z [x]
denote thering generated
over Zby
all ofthe values
x(a),
a EZ,
then it can be shown thatfxB,,,x
must be inZ[x]
for each n >0,
whenever x #
1.Thus,
as elements ofQ[X],
each suchBn,x
can be described ashaving
a denominator that is a divisor off x.
The casex =1 is best considered in terms of the classical Bernoulli
polynomials.
The
generating
function for the classical Bernoullipolynomials,
isgiven
by
and from the classical Bernoulli
polynomials
we obtain the classicalBernoulli
numbers, Bn
= The Bernoullinumbers, Bn,
areratio-nal
numbers, and, by
the von Staudt-Clausentheorem,
whenever0,
they satisfy
-Therefore,
whenever0,
the denominator of consists of theproduct
of thoseprimes
p such that(p -1) ~ n.
The classical Bernoulli
polynomials
are related to thegeneralized
Bernoullipolynomials
inthat, for X
= 1,
we haveBn,l(t) =
for all n, so also =
Therefore,
from the abovediscussion,
we have adescription
of thepossible
contents of the denominators of the classical and thegeneralized
Bernoulli numbers.By considering
congruences on the classical and thegeneralized
Bernoullinumbers,
wegain
information about their numerators. One of the morenotable
examples
of this is Kummer’s congruence for the classical Bernoullinumbers,
which states thatwhere c E Z is
positive,
with c = 0(mod p --1),
and n E Z ispositive,
even,
and n ~
0(mod
p -1) (see
[11],
p.61).
Note that we areusing
0~
to denote the forward differenceoperator, Acxn
= Xn. Moregenerally,
it can be shown thatwhere k E Z is
positive,
and c and n are asabove,
but with n > k.The
application
of Kummer’s congruence to thegeneralized
Bernoulli numbers was first treatedby
Carlitz in~1~,
with the result thatfor
positive
c EZ,
with c == 0(mod p -1),
n, k E Z with n > k >1,
andx such that
p~,
where p EZ,
p>
0. From[5]
(see
also[10]),
we seethat,
if theoperator
Aj
isapplied
to thequantity
then the congruence will still hold if the restriction n > k is
dropped,
requiring only
thatn >_
1. In addition tothis,
thedivisibility requirements
on c can be
removed,
yielding
a congruence of the form E for c, n, k EZ,
eachpositive, and X
such that EZ,
p~
0.Here,
we are
using
x~ torepresent
theprimitive
Dirichlet character withc~ the Teichmüller
character,
and we aretaking
q = 4 ifp =
2,
andq = p
otherwise. The values arise in connection with the
p-adic
L-functionLp(s; X)
of Kubota andLeopoldt
[8],
in thatthey yield
the values of thisfunction at the
nonpositive integers,
Lp(1-
n; X) =
/3n,x.
Withrespect
to this
expression,
the restriction c - 0(mod p -1),
as in theprevious
versions of Kummer’s congruence, causes each of x~ = Xn+c = ... =
when
p >
3.Thus,
theexpression
involvesgeneralized
Bernoulli numbers associated withonly
one Dirichletcharacter,
Xn.As an extension of the Kummer congruences, Gunaratne
(see [6], [7])
hasshown that
if p > 5, c, n, k E Z
arepositive, and X
=wh,
where h E Z andh
0(mod
p -1 ),
then EZp,
with a value modulopZp
that is c pindependent
of rn, andfor
positive
k’ E Z with k = k’(mod p -1).
Additionally, by
means of thebinomial coefficient
operator
where r E Z is
positive,
for these x, thesatisfy
‘ p r 1 °~ l ~3~,X
EZp,
also with avalue,
modulopZp,
that isindependent
of n.Young
[12]
hasextended these congruences to show that whenever c - 0
(mod
0(p’)),
for N apositive
integer,
we must EZp[X]
for allprimes
p and for all nontrivialprimitive
characters x, with = 1.Extending
work in[4],
we derive a collection of congruencesconcerning
the
generalized
Bernoullipolynomials.
These congruences are similar to the results ofGunaratne,
butthey
are without any restriction oneither X
or p.Given X
aprimitive
Dirichlet character and p aprime,
we considercongruences for the
polynomials
thatsatisfy
thegenerating
functionso that
(3n,x.
The make up the set of values taken onby
aparticular
two-variablep-adic L-function,
found in[4],
as thevariables ranges over the
nonpositive integers:
Lp ( 1-
~,;~) =
Since is a
polynomial
for each n, it follows thatas t -~
0,
in thep-adic
metric.Therefore, given
someposi-tive a E
Z,
there must exist aregion
R cCp,
containing
0,
such that forany t
E R we haveThus,
for certain valuesof
t,
we canexpect
to have the samemagnitude,
p-adically,
asthat of Our main result relates congruences on values of the
(3n,x(t)
in thefollowing
manner:Theorem 1. Let n, c,
k,
r EZ,
with n, cpositive
andk,
rnonnegative,
andlet
Fo
E Z be the smallestpositive multiple of
each... ,
Furthermore,
let t ECp,
withItlp
~
1~
and T EZp
suchthat Then
(a)
thequantity
E(b)
if
n’ E Z suchthat n’
> n, then(c)
if
k ispositive,
andif
c~ E Z ispositive
with c’54
c, then(d)
if
k’ E Z such that k’ > k and k’ - k(mod
~(~vN)),
for
somepositive
N E Z,
andif t
EZp,
then(e)
if t
EZp,
then8n,X (t))
E(f)
if
n’ E Z such that n’ > n, andif
t EZp,
thenMost of these congruences are extensions of the work of Guneratne. The
independence
of +T) - /3,~,x (t) ),
withrepect
to the value of n, when considered moduloqZp [X,
t],
isimplied by
(b).
Similarly,
(f)
implies
that thequantity
isindependent
ofn when considered modulo
qZp[x].
On the otherhand,
(c)
implies
that wehave
independence
withrespect
to the value of c when +T) -,Q~~X(t))
is considered modulot~.
There are no resultsinvolving
independence
withrespect
to c in any of the earlier works[6], [7], [12],
or[13],
so that this appears to be a new direction ofinvestigation
withrespect
to Kummer’s congruence whenapplied
to thegeneralized
Bernoullipolynomials
and numbers.Both
(e)
and(f)
in the above can begeneralized
quite
nicely.
As wehave seen, the binomial coefficient
yields
apolynomial
in
having
coefficients inQ.
Thus,
togeneralize
(e),
one maywish to consider what other
polynomials
in whenapplied
to!3n,x(t
+T) -
!3n,x(t),
yield
values inZp [X].
Thefollowing result,
which canbe found in
[9],
enables such ageneralization:
Theorem 2. A
polynomials
f (x)
EQp[x]
of degree
N mapsZp
intoZp
if
and
only i f it
can be written in theform
where
bo, bl, ... , bN
EZp.
Assuming
thehypotheses
of Theorem1,
along
with the additionalrestric-tion that t E
Zp,
since +T) -
E for eachnonnegative
r EZ,
any sum of the formwhere
bo,
bl, ... ,
bN E
Zp,
must also be inZp [X] -
Thus,
Theorem 2implies
that every
polynomial
EQp [x]
that mapsZp
intoZp
yields
anoperator
such that
I /
It then
follows, by
(f),
thatLetting
t = 0 in Theorem 1yields
congruences on~i~,x (T) - /?~(0),
and, thus, provides equivalences
between congruences on/3n,x(r),
involving
generalized
Bernoullipolynomials,
and congruences on/3n,x(O),
involving
generalized
Bernoulli numbers.Furthermore,
if n is even andx ( -1 ) _ -1,
or if n is odd andx(-1)
=1, X =f:.
1,
then =0,
with theonly exception
for the case X =1
being
when n= 1,
for which we haveBl,l = 1/2.
Thus,
given
x, we canalways
find n and c such that =0,
implying
that each of the above congruences hold for the
corresponding
provided
we have theappropriate
restrictions on T.A
polynomial
structure for the classical Bernoullipolynomials,
similar to that of,8n,x (t),
isincorporated
in a result of Eie andOng.
In[3],
they
apply
Kummer’s congruence to showthat,
forp > 5,
if n, c E Z are eachpositive
with n fl
0(mod
0(p))
and c - 0(mod 0(p)),
thenwhere T > 0 is an element of
Q
having
denominatorrelatively
prime
to p,and j
E Z with both0 j p -1
and j
pT. This has been extended toapplications
ofAk,
with k >1,
by Young. Furthermore, Young
shows that similar congruences hold for thegeneralized
Bernoullipolynomials
associated with certainprimitive
Dirichlet characters[13].
We continue ourwork
by
showing
how the congruences of Eie andOng
with1 j p -1
relate to the same congruenceswith j
= 0.Finally,
we consider theexpression
where t E
Cp, Itip
1,
and discuss thestrength
of the congruences thatcan be obtained with
regard
to it.2. Preliminaries
One of the more well-known
properties
of thegeneralized
Bernoullipoly-nomials is
that,
forpositive
m EZ,
for all n > 0. This can be derived from
(1).
Similarly,
for the classicalBernoulli
polynomials,
(2)
implies
that,
whenever m E Z ispositive,
for all n > 0. These
properties
arekey
to the derivation of our congruences, as we will see.Let p be a fixed
prime.
We will useZp
torepresent
the set ofp-adic
integers,
andQp
the set ofp-adic
rationals. LetCp
denote thecompletion
of thealgebraic
closure ofQp
under thep-adic
absolutevalue 1.lp,
normalizedso that
lplp
=p-1.
Notethat,
for each t ECp,
there exists some a EQ
such thatitlp
= Fix anembedding
of thealgebraic
closure ofQ
intoCp.
Each value of a Dirichletcharacter X
is either 0 or a root ofunity.
Thus,
we may consider the valuesof X
aslying
inCp.
Denote
q = 4 if p = 2
andq = p if p > 3.
Let wrepresent
the Dirichlet characterhaving
conductorfw
= q, and whose values EZp
satisfy
= 1 and
cd(a) -
a(mod
qZp)
for(a, p)
=1,
with = 0other-wise. For an
arbitrary
character x, we then define the character xn =Xw-n,
where n E Z,
in the sense of theproduct
of characters.Let
(a)
=w-1 (a)a
whenever(a, p)
= 1. We then have(a~ -
1(mod
qZp)
for these values of a. For our purposes we extend this notation
by
defining
(a+qt)
= for all a EZ,
with(a, p)
=1,
and t ECp
such thatItlp
1.Thus,
(a-f-qt) _
so that(a + qt) =-
1(mod
qZp(t~).
We now use these
quantities
to express the differencefor F E Z a
positive multiple
ofpq-1 fXn’
in a manner that will enable theproof
of the main result.Lemma 3. Let t E
Cpi
ltlp
1,
and let n E Z bepositive.
Then, for
F E Z a
positive
multiple of
pq- 1 fXn
IThus, by
(3),
we can writeSince
xn(a)
= and +qt) _
(a
+qt),
whenever(a, ~)
=1,
the result follows. 0Lemma 4. Let i and m be
integers
such that 2 i m. Then(7)qpi-2 =
0
(mod
m(m -
1)PZp).
Proof.
For 2 i m,Since
(i, i
-1) =
1,
the lemma will follow if we show that0
(mod
pZp)
andqpi-2/(i - 1) ==
0(mod
pZp),
whenever i > 2.If p
=2,
then the powerof p
that dividesany j
EZ,1 j
i,
is at mosti/2.
Thus,
the powerof p
that dividesqpi-2 / j
is at least i -i/2
=i/2.
Fori >
2,
thisimplies
that the powerof p
that dividesqpi-21j
is at least 1.Therefore,
qpi-2/ j -
0(mod
pZp).
If p > 2,
then the powerof p
thatdivides j
EZ, 1 j i,
is at mosti/3,
so the powerof p
that dividesqpi-2/ j
is at least i - 1 -z/3
=2i/3 -1.
For i >
2,
the power of p that divides is then at least1/3,
and,
thus,
at least 1.Therefore,
0(mod
pZp).
Therefore,
for anyprime
p, thequantity
0(mod
pZp)
when-ever j
E Z with1 j
i,
implying
the result. 0In the
following
lemma,
we derive congruenceproperties
of thequantity
(a
+ that will be utilized to prove the main result.Lemma 5. Let a, m E
Z,
wath(a, p)
= 1 and m >2,
and let t ECp
such thatltlp
1. ThenProof.
Recall that(a
+qt) = (a)
+qw-1(a)t.
Since(a)
EZp
andThus,
Lemma 4 then
implies
thatThe result follows since
qb(a) =
(a) -
1.As a weaker form of this
lemma,
we have the congruencewith the same restrictions on a and t as in the
lemma,
but for allpositive
m E Z.
For each r E
Z,
r >0,
thequantity
(jj)
is defined in like manner as thebinomial
coefficients,
denoting
(~)
= 1 andfor r > 0. Note that each such
quantity
is apolynomial
in x, and has theexpansion
where the values
s(r, m)
areStirling
numbers of the firstkind,
definedby
the
generating
functionThe
s (r, m)
areintegers,
where r, m EZ,
r >0,
m >0,
satisfying
s (r, m) =
0 whenever 0 r m, and
s(r, r) =
1 for all r > 0. For additionalinformation on
Stirling
numbers of the firstkind,
we refer the reader to[2],
pp. 214-217.
3.
Congruences
forgeneralized
Bernoullipolynomials
From Lemma 3 we have an indication that thedivisibility properties
of
(3n,x(t
+T) -
(3n;x(t)
must be similar to thedivisibility properties
of(a
+ whenever a E Z with(a, p)
= 1. We will now consider how wecan utilize this to derive a collection of congruences
concerning
theoperator,
xn.Repeated application
of thisoperator
can beexpressed
in the formfor some
positive
k E Z.We can now prove our main result:
Proof of
Theorem The set ofpositive integers
inFoZ
is dense inFoZp.
Thus,
for T EFo Zp,
there exists a sequence inwith Ti
> 0 for eachi,
such that Ti ~ T. Thisimplies
that anyf (x)
E mustsatisfy
~f (~),
and, thus,
any congruence satisfiedby
each must also be satisfiedby
f (T) .
Since each of0
(3,x
(t
+x) -
6,,,x (t»
andc
are in for t inCp,
any of the statedcongruences that hold for all
positive
T EFoZ
must also hold forarbitrary
T ~
FoZp*
(a)
SinceAc
is a linearoperator,
Lemma 3implies
thatwhere F is a
positive multiple
ofFo.
Note thatso
that,
By
Lemma5,
((a
+qt)~ -
1)k =
0(mod
implying
the result.Lemma 5
implies
thatand the result follows.
(c)
By utilizing
(7),
we obtainwhere F is a
positive
multiple
ofFo.
For each a E
Z,
with(a, p)
= 1,
Lemma 5implies
thatwhere u, v E
Zp [t],
eachdepending
on a. Inparticular,
u =q-1((a) - 1)
+úJ-1(a)t.
Therefore,
Similarly,
there exists v’ EZp[t]
such thatNow consider
This can be
expanded
toyield
By
Lemma4,
(m) prn -
0(mod
kp)
for each m > 1.Also,
for these samevalues of m, the
quantity
(c -1)mvm - (c’
is 0 modulo(c -1,
c’(d)
From(7)
we can writewhere F is a
positive
multiple
ofFo . Now,
Lemma 5implies
that1= 0
(mod
cqZp).
Thus,
if a is such that(a
+qt} -1- 0 (mod
then
However,
if a is such that(a
+(mod
thensince k’ -
k - 0(mod
~(p~)).
Therefore,
the result.(e)
We are onceagain working
with a linearoperator,
so Lemma 3implies
thatwhere F is a
positive multiple
ofFo.
Utilizing
(5),
and then(6),
we canwrite
Therefore,
From Lemma
5,
EZp
for each a E Z with(a, p)
=1,
implying
the result.(f)
Whenever F is apositive
multiple
ofFo ,
(8)
implies
thatThe
quantity
(a
+qt)n’-n -1
must be 0 modulo(r~’ -
n)qZp,
by
Lemma5,
so the result follows. 0
Note that both
(e)
and(f)
require
that t EZp.
This is because thequantity
c-kq-k((a
+ EZp
if this istrue,
and, hence,
I I
for all
integers
r >0,
whereas,
this will not hold for all values of r forcertain t E
Cp.
However, if r p,
then(p, r!)
=1,
so thatfor any t E
Cp
withltlp
1.Thus,
we can still obtain congruences thatcorrespond
to those of(e)
and(f)
for such values of t whenever r p.4. The classical Bernoulli
polynomials
and the congruences of Eie andOng
We will now show how the congruences that we derived for the
classical Bernoulli
polynomials.
For eachpositive n
EZ,
denoteNote that this structure involves
only
the classical Bernoullipolynomials.
Consider how to relate with
/3n(t).
Given we need X = w’~in order for x~ =1.
Thus,
Since
B~(-t) _
+ for all n >0,
the aboveexpres-sion
simplifies
toso
that,
from Lemma3,
where F E Z is
positive.
Forpositive
c E Z such thatO(q) I
c, we must have =o/B
so thatwhich is a
quantity expressed solely
in terms of the classical Bernoullipolynomials. Therefore,
some of the congruences of Theorem1,
involv-ing
!3n;x (t)
with X = will also hold for!3n (t),
provided
~(q) ~
c.
Infact,
since our character wn does not vary with c or
k,
each of(a), (c), (d),
and(e)
of Theorem 1 will hold for congruences on!3n(t + T) - #n(t),
with the same restrictions on c,k,
n,t,
and T,except
also with§(q) c.
Note that our valueFo
= 1.However,
in the cases(b)
and(f),
thecorresponding
characters,
wn andwn~,
are not the sameunless n’ - n
(mod ~(q)).
Thus,
under the
hypotheses
of each of(b)
and(f),
and theresulting
generaliza-tion of
(f),
along
with the restriction~(q) ~ c,
theresulting
congruences of these cases can be obtained for the difference,~3,~ (t + T) - !3n (t)
provided
weinclude the additional restriction that n’ = n
(mod ~(q)).
Now let t E
Cp,
ltlp
1,
andlet j
E Z with1 j
p - 1. Eie andOng
obtained congruences on anexpression
of the formWe will see how congruences on
/3~~~ (t)
relate to congruences on!3n(t).
By
utilizing
(4),
we can writeSince
1 j p -1,
we must also have(a, p)
= 1 for 1 aj,
so thatThe congruences on the difference +
r) -
were derived becauseof certain congruence
properties
that are satisfiedby
cvn(a) (a
+qt)n.
Thisquantity
is very similar toWn(-a)(a - qt)n,
so that the congruences thatwere obtained with
regard
to this difference should findparallels
incongru-ences on
,(i~~~ (t)
-/3n (t),
involving
theexpression
of Eie andOng,
provided
~(q) ~ c.
Thefollowing
result,
concerning
the difference,Q~~~ (t)
-/3,~(t),
canthen be obtained:
Theorem 6. Let
n, c, k, r
EZ,
with n, cpositive,
c - 0(mod ~(q)),
andk,
rnonnegative.
Furthermore,
let t ECp,
withItlp
1. Thenfor j
EZ,
with
(a)
thequantity
EZp[t] ;
(f)
if
n’
E Z such thatn’
> n and n’ - n(mod
Ø(q»,
andif
t EZp,
thenProof.
We derive theidentity
for the
application
of the forward differenceoperator,
and for the binomial coefficientoperator
The congruences then follow from the
analysis
of theseexpressions by
thesame means used in the
proof
of Theorem 1. DAssuming
thehypotheses
of Theorem6,
along
with the restriction thatt E
Zp,
Theorem 2 can also beapplied
to thequantity
~3~~~ (t) -
toyield generalizations
of(e)
and(f). If f (x)
E mapsZp
intoZp,
thenis an
operator
for whichFurthermore,
, r ,
where n’
E Z suchthat n’
> n and n’ =- n
(mod
5. Additional congruences for the case p = 2
We now wish to consider congruences on the
expression
taken over
Cp,
with t ECp,
ltlp
1. Notethat,
forprimes
p >
3,
thisthe
only
additional interest in congruences on thisexpression
occurs whenp =
2,
the case for which we now restrict theremaining
discussion.In the case p =
2,
we write the differencewhere n E Z is
positive, F
E Z is apositive
multiple
of and t EC2
with
ltl2
1. Thequantity
(a)
+ does not havequite
the samecongruence
properties
as(a
+4t) = (a)
+4c~-1 (a)t,
and our congruences on thisexpression
donot,
ingeneral,
attain the samestrength
as those on(a
+4t),
as in Lemma 5.Instead,
we have thefollowing:
Lemma 7. Let a, m E
Z,
with(a,
2)
= 1 and m >2,
and tet t EC2
such thatltl2
1.Farthermore,
letto
EC2,
ItOI2
=1,
and a EQ, a >
0,
such that t =2"to.
ThenThe
proof
of this result can be obtainedby
the same means that weutilized to derive the
proof
of Lemma 5. As a weaker form of thislemma,
we havewith the same restrictions on a and
t,
but for allpositive
m E Z.From Lemma 7 we can then derive congruences similar to those found in Theorem
1,
having
the samestrength
as those results if1212,
butweaker if otherwise. Results similar to those of Theorem 6 can also be
derived. The methods
given
in theproof
of Theorem 1 can beemployed
inthese
derivations,
which we leave to the reader.Acknowledgements
Finally,
the author would like to express hisgratitude
to hisgraduate
advisor,
Professor Andrew Granville of theUniversity
ofGeorgia,
Athens,
forsuggesting
the consideration of thisproblem.
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Glenn J. Fox
Department of Mathematics and Computer Science
Emory University Atlanta, GA 30322, USA E-mail : foxGmathcs .emory .edu