Congruences for Bernoulli numbers and Bernoulli polynomials

DISCRETE MATHEMATICS Discrete Mathematics 163 (1997) 153-163

ELSEVIER

Congruences for Bernoulli numbers and Bernoulli polynomials Z h i - H o n g Sun

Departmentof Mathematics,Haaiyin TeachersCollege,Huaiyin,Jiangsu223001, China Received 4 January 1995;revised 20 July 1995

Abstract

Let {B.(x)} be the well-known Bernoulli polynemials. It is the purpose of this paper to determine pB~p- t~+b(x)modp ", where p is a prime, k, b nonnegative integers and x a rational p-integer.

1. Introduction

The Bernoulli numbers {Bn} and Bernoulli ~:olynomials {B.(x)} are given by y.(n1 n) Bo=l,

~_,o\k Bk=O

(n=2,3,4 .... )

and

Bn(x)= ~ (:)Bn-hx ~. k=O\

/

It is interesting to investigate arithmetic properties of {B,} and {Bn(x)}. For the work on this line one may consult [1-3, 5-9, 11, 12]. Here we give two classical results (cf. [83): Kummer's congruences: Let p be an odd prime, and b an even number with p - 1 Y b. For k = 0.1, 2 . . . . we have B~p-l)+b

k(p-l)+b-

_ Bb (modp).

b

Von Staudt-Clausen Theorem: Suppose that p is a prime and k ¢ Z +. Then ~0(mod p)

/f p --- 2 and k > 1 is odd,

pBk(p- 1) -- ~ _ 1 (mod p) otherwise. 0012-365X/97/$17.00 © 1997Elsevier Science B.V.All rights reserved SSD! 0012-365X(95)0031 6-9

Zhi-Hong Sun / Discrete Mathematics 163 (! 997) ! 53 - 163

154

In [10] the author proved that pBk~p-l~ = kpBp-l - (k - 1)(p - 1) (modp2),

pBk~p-l~--

(k)

2 pB2~-2-k(k-2)pBp-1

+

(k-l) 2

( p - 1) (modp3),

where p > 3 is a prime and k ~ 7/+. Let p be a prime, and 7/p the set of rational p-integers. Suppose that b ~ 7/+ w{0} and x ~ 7/p. In this paper we determine PBkcp-1~+b(X)modp" by proving that

-

pBk,p_l)+b(x)pk(p_l)+bBk(p -

=

(--1)"-l-r

l~+b (x

(: 11

q- ~ : X ) p )

PBr{p-l)+b(g)--Prtl'-l)+b

r=o

= a n k " + a n - l k " - 1 + ... + a l k + a o

(modp n) (k=O, 1,2 .... ),

where ao, a~. . . . . an are all integers, < - x > p is the lea-~t nonnegative residue of - x mod p, and 6(n,b,p)=~l

if Bn¢7/p and p - l i b ,

(1.1)

{o ifBn~Zp or p - l A ' b . Clearly, our results provide a wide generalization of yon Staudt-Clausen theorem.

2. Basic lemmas In this section we give several lemmas which will be used later. Lemma 2.1. Let n >i 1 and k >t 0 be integers. For any function f we have .-i

1 --;

kr f(r)

where the second sum is zero when k < n.

'

1)'~. (r~(_l)Sf(s),

155

Zhi-Hong Sun~DiscreteMathematics 163 (1997) 153-163

Proof. Note that

( : ) . 1- ~_ ,, = o ( _ 1 ) , (X) - l r

.

=~o(_1),,= x

we find

~z ( k ~ ( - 1 ) ' E ( r ~ .( - 1 ) S f ( s ) - - s ~ _ _ o ~.,~, =s( ,=o\r/

s=okS/

k

)()

r

r (-l)'-Sf(s)

s

.g(k :),_,~-..,s,_: ,~o,,, ~o,, ; .~o z(':).~.,,

/),_l.,.,

-I k (s __.~,_l,.-, .(:-, 1 - - ss)(,),,

Now applying binomial inversion formula fields the result.

[]

Lemma 2.2 (Raabe's theorem (cf. Ireland and Rosen [8])). Let

m, n ~ Z + a n d x ¢ R.

Then

"-'

m'-' Y.,.(x + r=O

~) =,.(m~).

\

Lemma 2.3. Suppose that p > 1 and k >>.I are integers. I f x, Xo ¢ Zp and x =- Xo (mod p) then

pe,,(x), e,,(x)-~ e,,(Xo), e~(x)_k_e%z'' ProoL For n >t 1 we have pBn, p~-~/n~Zp. Thus, k

.=okr/ It is well known that Bh(xo + t) =

~(~)

B,-,(Xo)f.

r=O \

/

156

Zhi-Hong Sun~Discrete Mathematics 163 (1997) 153-163

So

r=l

=

~ / k - 1\ ,=,

~,

_,

/ x - Xo\" p'- 1

) ~"~-,~-~o, ~ - - ~ )

• - 7 - ~ ~,.

Assume x-~ n(modp) for some n ~ Z +. By the above and the fact Y.,~__-olr ~-z (Bk(n) -- Bk)/k we obtain B~(x) - Bk k

Bk(x) -- B~(n) Bk(n) -- B k 7/p, k + k

which completes the proof.

[]

3. Congruences of the Kummer type In this section we prove the following. Theorem 3.1. Let p be a prime, n e e +, b~7/+u{0} and x ~ g p . Then

fO(modp") =[p"-t(modp")

if n ~ S o or p - 1Xb, if n¢Sp and p - l i b ,

where Sp = {mlBm~Ep, m~7/+}

= f{3,5,7 .... } [{mlp-lXm, and ( - x ) p

if p = 2 , m e E +} / f p > 2

denotes the least nonnegative residue of - x modulo p.

Proof. Since B.(x) = ~ ( n ~ B , x " - " ,=o\r/ one has

Zhi-HongSun~DiscreteMathematics163(1997)153-163

157

Using Raabe's theorem one sees that

p-k'o-')-b+'B.(p_l)+b(X)--Bk(p_l)+b(.x'l-~- --X>p)

j=o

P

="P~+b(k(p-l)+b~ ,=o

\

v

/

"~ (x+J~"'-"+b-'B. ' /

/~o> .

r

.

which gives

(x + <-x>,'~

pBk~p-l)+b(X)-- ph~p-l)+aBk~p- l)+b P / "P-"+b(k(p-1)+b~ "£' (x+j),,p-,,+b-.p.B,. = 2r=O r il jgo ) •

-x

p

It then follows that

~o(:)(--1)'(pnk,p_i)+b~X)--okgp-1)+'B,(p_,,+b(X'l-~-~--x>:~)) n Mp-l)*b/ n\ ( 1)ark(p-- 1) -~-b~ =k~=o ,~=0 Ik) - ~ ) j;;O>o P~ (X +Jf'P-"÷b-'P'B" r n(p-l)+b ) ---- ,=o ~'~ P'B" .~=o(k'-' j~=o " n)(-l~'( k`p-rl)+b (x+J)a'P-U+b-" j~<- )~ Let f(k) be a polynomial of degree r with the propertyf(k)~ Z for k ~ Z. It is well known that f(k) = ~'=oa.d~) for some ao. . . . . a.~Z. Now, tettingf(k) = ( ~ - lJ+b) we get ( _ 1)~ n

•

(

k(p

)

1) + b (x + j ) ~ P - l ) + b - ,

-

r

n

_-

-

.

----'~=o(--l)~a.(:)~(--l)'-'(nk--:)(X+J)k'P-t'÷b-" (observing that

(:)(~)= (:)(; ~ :))

-- ~ ( - 1)Sa~(n~(x +j)~IP-U+b-'(I--(x + j ~ - ' ) " - ' . • =o ks/

Zhi-Hong Sun~Discrete Mathematics 163 (1997) 153-163

158 So

n

k=O

+

k

,=o

,¢;%2=o

Since

p'B,.( l -- (x .+-j)t'- l)"-s -- p"-~+'- l.pB .(.l -- (x ; j)p- l),-~ forj ¢ (-x>p

-

O(mod

p")

a n d s ~ { O , 1. . . . . r - 1} w e see t h a t

,=o

j2<%,

(:)

( ; )

-x>p ( u s i n g t h e fact a , = ( p - 1)" - ( - 1)" ( r o o d p~ a n d F e r m a t ' s little t h e o r e m )

~-

{X .}_j)b-np, • j=o J # < x>p

J

# < -'Or

j

2(2°>,

P'I') B, ((. ~

,=o\r/

+j) l - ( x +j)P- P~"-'

X,

p

P

p-I

- p"B, ~_~ Ix+j) t'-~ j=c j # < -~>,

•

P

B , ~ arv b y L e m m a

/

Zhi-Hong Sun/DiscreteMathematics 163 (1997)153- 163

if B,E&,

O(modp”) p-1

p”-‘(P-l)

~

I.@=

s=1

p”-‘(modp”)

if B.$Z,

and p - 1 lb,

if B.$Z,

and p - 1,fb.

p-1

p”-‘(p - 1) 1

&’ E O(modp”)

s=1

(Note that pBR = p - 1 (mod p) and that p - 1 In if B,$ Z,.) This completes the proof.

0

Corollary 3.1. Suppose that p is a prime and x E Z,. If p i n - b +

!

and b > 0 then

+n (-l)kpB~~p-,,+~(x)-pbBb~~)+~(n,b.P)~-l k&o0 k

@odd”),

wkere 6(n, b, p) is defined by (1.1).

Proof. Since k(p - 1) + b > n for k > 0 we have

; co n

k=O

(-l)k$‘p-“+b&,p-l,+b

(E!$G)

This together with Theorem 3.1 gives the resuit.

E

pbBb (C$S)

(mdp”).

IJ

4. Congruences for pB,,-,,+,(x)nmd#’ Theorem 4.1. Let p be a prime and XEZ,. ao,

... ,a, E Z, PBk,,-

For each positive integer n there are

such that

l,+b(X)

=aa,kn+

-

PktP-l)+bBk(p-lj+b

. . . + alk + ao(modp”)

for every k = 0, 1,2, ... . Furthermore, ifp > n then ao, Q~,. . . , a,(mod p”) are uniquely determined; ifp - t ,f’ b or B,, E BP we may assume a, z O(mod p”).

Proof.

set

Zhi-HongSun / DiscreteMathematics 163 (I 997) 153- 163

160

Then Ao = p(pBb(x) - pbBb(X + ~-x>P)). lt followsfrom Theorem 3.1that An~Zp for n >/1. Thus, applying binomial inversion f.,-rmula we obtain

,=o\r/ = pB~(x)-- pbBb(:x + (~- x > e ) p,- t k(k- l)--.(k- r + 1)(- l)'-rTA,(modpn).

+

(4.1)

r=l

Since p'-l/r!eT/p for r >t 1 we have

pBk,p_ ,,+b(x) -- pk'p- t'+l'Bk,t,_ t,÷b(x + ~ -x)t'-) -- ank n + .-. + a~k + ao(modp") for some ao, a~. . . . . anE7/p. Let {S(n, k)} be the second kind Stirling numbers. F r o m [13, 4i we know that

~ ' ~ ( m ~ ( _ l ) , _ , r , : { o , [ S ( n , m ) if n>~m, ,=o\r/ if n < m. Hence,

r=O r=O k=O = (-- l)'%l!(a m + ... + S(n, m)an) (m = O, 1. . . . . n). (Note that S(m, m) = 1.) Forp>nwehavem
PBk,r ..... (x) -- Pk't'- " +hBk,t,- l , +b (7~C+ (-fi--X >t~) = ~ r=o

( - - l ) ' p ' - t A , - = ~ a,k"

(modp")

r=O

for some ao . . . . . an- t ~ 77p. Now. clearly we may assume an -- 0(modpn). Hence, the theorem is proved.

[]

Zhi-HongSun~DiscreteMathematics163(1997)153-163

161

Remark 4.1. Using the properties of Stifling numbers one can choose ao . . . . . a~ so that as. s!/ps- 1~ T_pfor every s = 0,1 . . . . . n. As examples we point out the following congruences: (2 - 22k)B2k -- 12.(2k) 2 + 18-2k + 1 (mod27).

(4.2)

(3 -- 32k)B2h ------ 36k 4 + 1OSk3 - 93k 2 + 18k + 2 (mod 3s).

(4.3)

(5 - 54k)B4k :- -- 125k 4 + 125k a - 300k 2 - 100k + 4 (mod 54).

(4.4)

We remark that F r a m e [6] has given a congruence for 30B2n rood 211.35 "56. F r o m the congruence he derived that 3 0 B 2 n - 1 + 6 0 0 0 ( n - )12

(m°d 2a" 33" 53)"

Corollary 4.1. Let p be an odd prime and x ~ Zp. If k, n and b are integers with k, n > 0 and b >10 then

pBk~,f,+b(X)= pBb(x)-- pbBb(X "l" ? x ) ' )

(modf),

where ~pdenotes Euler'sfunction. Proof. F r o m Theorem 3.1 we have

j~=oC )(-- I)J(PBj~p-,,+b(X)-- f P - "+bBj,p- ,,+b(X + ( ; X~P)) -- O(mod p)• Observe that p'- l/r! --- 0 ( m o d p ) for r > I and that kq~(f) + b >1 f - l(p _ 1) > n, by using (4.1) we obtain the result. Remark 4.2. In [1, 2], Carlitz proved Corollary 4.1 in the case b = 0. When x -- 0 and p - 1 X b our result can be deduced from K u m m e f s congruences (cf. [8,13]). We also remark that the congruence holds for p = 2 if n > 2. Theorem 4.2.

Let p be a prime, k, b ~ Z + u { 0 } and x~Z r For n =

1,2,3 . . . .

we have

pBa,p- ,,+b(x) -- p"P- l'+bB,,p- l,+b(X "t-

p) "-'

( - 1 - r~ fk~ f

-- ~ (-- I)"- ' -" \~ r=O

1--r/~r)~ pB'cp- ,/+b(X)

-- p'W-|"~B,~p_l,+b~ + <;X>P)) + (--l)"~(n,b,p)(:)f -1

(rood f'),

!62

Zhi-HongSun/ Discrete Mathematics 163 ( ~997) 153-163

where 6(n'b'P)~-{lo

if Bn ~ 7/p and p - 11 b, otherwise.

Proof. This is immediate from Theorem 3.1 and Lemma 2.1. [] Remark 4.3. When p - 1 ;l" b we will prove in another paper the following stronger congruences: Bk(p_l)+b(X)-- pk(p-l)+b-lBk(p_l)+b(X d- ? X~p') k(p - 1) + b ~--~(_i)._i_ ,

,=o

-- I --r

k

1

r

r

Br(p - 1)+b(x ) - - P_,(p- l)+b- i Br(p- ,)+b(X + ( ; X) ) p X

r(p -- 1) + b n-I - ) ' a , k " (modp") (ao, al . . . . . a , - l e Z ) , r=o which is a wide generalization of Kummer's congruences. 4.2. Let p be a prime, and k, n, b be three integers with k >>.O, b >>.O, n >1 1.

Corollary

Then (l -- p'~P- "+b- ')pBk,p_ l)+b -- n ~ (-- l)"-1-'(kn -- l -- r) ( k ) ,=o l -- r r x(l __pr(p-l)+b-l)pBr(p - l)+b .a¢.(__|)n~(~, b,p)(kn)Pn-I Corollary

(modpn).

4.3. Suppose k, n ~ Z + and x ~ ~-2. Then

2Bk(x)

-

2kBk(:,¢ + ( f X~2~ n - , k l-r ,, -/-~o(-l)"-'-'(n-l-r)(r)

x(2B,(x)-2"B,(X+~X)2))+(kn)2nB,

k

(mod 2~).

Acknowledgement

I am grat,=fui to the referee for his suggestion on abridging the proof of Theorem 3.1.

Zhi-Hong Sun / Discrete Mathematics i 63 (! 997) ! 53 - ! 63

! 63

References I'll L. Carlitz, A divisibility property of the Bernoulli polynomials, Pro(:. Amer. Math. Soc. 3 (1952) 604-60?. [2] L. Carlitz, Some congruences for the Bernoulli numbers, Amer. J. Math. 75 (1953) 163-172. ['3] L. Carlitz, Kummer's congruence for the Bernoulli numbers, Portugal. Math. 19 (1960) 203-210. [4] D.I.A. Cohen, Basic Techniques of Combinatorial Theory (Springer, New York, 1978) ! 19. ['5] K. Dilcher, L. Skula and 1. Sh. Slavutskii, Bernoulli Numbers Bibliography (1713-1990). [6] J.S. Frame, Bernoulli numbers modulo 27000, Amer. Math. Monthly 68 (1961) 87-95. [7] F.S. Gilllespie, A generalization of Kummer's congruences and related results, Fibonacci Quart. 30 (1992) 349-367. [8] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory (Springer, New York, 1982) 228-248. [9] H.R. Stevens, Bernoulli numbers and Kummer's criterion, Fibonacci Quart. 24 (1986) 154-159. [10] Z.H. Sun, A note on Wilson's theorem and Wolstenholme's theorem, Proc. Amer. Math. See., submitted. I'll] H.S. Vandiver, Certain congruences involving the Bernoulli numbers, Duke Math. J. 5 (1939) 548-551. [12] H.S. Vandiver, On developments in an arithmetic theory oftbe Bernoulli and allied numbers, Scripta Math. 25 (1961) 273-303. [13] L.C. Washington, Introduction to Cyciotomic Fields (Springer, New York, 1982) 61-64.