Poly-Bernoulli numbers

M

ASANOBU

K

ANEKO

Poly-Bernoulli numbers

Journal de Théorie des Nombres de Bordeaux, tome

9, n

o

_{1 (1997),}

p. 221-228

<http://www.numdam.org/item?id=JTNB_1997__9_1_221_0>

© Université Bordeaux 1, 1997, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

Poly-Bernoulli

numbers

par MASANOBU KANEKO

RÉSUMÉ. Par le biais des séries

_{logarithmiques}

_multiples,

nous

définissons

_l’analogue

en

_plusieurs

variables des nombres de

Ber-noulli. Nous démontrons une formule

_explicite

ainsi _qu’un

théo-rème de dualité pour ces nombres. Nous donnons aussi un

théo-rème de _type von Staudt et une nouvelle preuve d’un théorème

de Vandiver.

ABSTRACT.

_{By using polylogarithm}

series,

we define

"poly-Ber-noulli numbers" which

_generalize

classical Bernoulli numbers. We derive an

_explicit

formula and a

_duality

theorem for these num-bers,

together

with a von

_Staudt-type

theorem for di-Bernoulli numbers and another

_proof

of a theorem of Vandiver.

For every

integer k,

we define a _sequence of rational numbers

_(n

=

0,1, 2, ~ ~ ~ ),

which we refer to as

_{poly-Bernoulli}

_numbers,

_by

Here,

for _any

_{integer k,}

_{Lik (z)}

denotes the formal _{power series}

_(for

the k-th

polylogarithm

if k &#x3E; 1 and a rational function if k

0)

When k =

1,

is the usual Bernoulli number

(with

Bi1)

=

1/2):

,..-v

and

_1,

the lefthand side of

₍₁₎

can be written in the form of

"iterated

_{integrals" :}

In this _paper, we

_give

both an

explicit

formula for

B~~~

in terms of the

Stirling

numbers of the second kind and a sort of

_duality

for

_negative

index

poly-Bernoulli

numbers. Both formulas are

_elementary,

and in fact almost

direct _consequences of the definition and

_properties

of the

_Stirling

numbers. As

_{applications,}

we _prove a von

_Staudt-type

theorem for di-Bernoulli

num-bers

_(I~

=

2)

and

give

_an alternative

proof

of _a theorem due to Vandiver on a _congruence for

1.

_Explicit

formula and

_duality

An

_explicit

formula for

_B~~~

is

_{given by}

the

_following:

THEOREM 1.

where

is the

_Stirling

number

_of

the second kind.

REMARK. When k =

1,

the theorem and its many variants are classical

results in the

_study

of Bernoulli numbers

_(cf.

_(1]).

Because the

_Stirling

numbers are

_integers,

we see from the formula that

for k 0 is an

_integer

(actually

_positive,

as demonstrated in the

remark at the end of this

_section).

THEOREM _{2. For any}

_{n, k &#x3E; 0,}

we have

PROOF OF THEOREMS 1 AND 2. One _way to define the

_Stirling

numbers of the second kind

_{S(n, m) (n &#x3E;}

_{0, 0}

m

_n)

is via the formula

where,

for each

_integer

m &#x3E;

_0,

we denote

by

the

_polynomial

x(x

-1)(x-2)

(-m+1)

((x)o

=

1).

Then

they

satisfy

the

following

formulas

(when n

= 0 in

(3),

the

identity

00

= 1 is

understood):

For other

_definitions,

further

_properties

and

_proofs,

we refer to

_[3].

Now Theorem 1 is

_readily

derived from the definition

₍₁₎

and the formula

(4).

In

_fact,

Hence the theorem follows.

To prove Theorem

2,

we calculate the

_generating

function of

B~ ’~~ :

The last

_expression

_{being symmetric}

in x

_{and y}

_yields

Theorem 2. REMARK. Since

2. Denominators of di-Bernoulli numbers

Using

Theorem

_1,

we can

_completely

determine the denominator of di-Bernoulli numbers as follows.

THEOREM 3.

(1)

When n is

odd,

B~2~ _ -~n42 Bnl~l

(n &#x3E;

1). (Hence

the

descrip-tion

_of

the denominator reduces to the classical Clausen-von Staudt

theorem.)

(2)

When n is even

(&#x3E; 2),

the

_p-order

ord(p, n) of

_for

a

prime

number _p is

_given

as

follows.

(a) ord(p, n) &#x3E;

0

_{if p}

&#x3E; n + 1.

(b)

we have:

(i) ord(p, n) =

-2

_{if p -}

(ii)

If p -

1

_An,

then:

(A)

0 or n - n’ mod

p(p - 1)

for

somme 1 n’ _{p - 1.}

(B) ord(p, n) =

-1 otherwise.

(c)

ord(3, n) &#x3E;

0

_{if n -}

2 mod 3 and n &#x3E; 2. Otherwise

_{ord(3, n) =}

-2.

(d) ord(2, n) &#x3E;

0

_{if n -}

2 mod 4 and n &#x3E; 2.

_{ord(2, n) =}

-1

_if

n - 0 mod 4.

_{ord(2,2) =}

-2.

Before

_proving

the

_theorem,

we establish the

_following

_lemma,

which will be needed in the

_proof.

LEMMA 1. Assume n &#x3E; 2 is even and

_{p &#x3E;}

5 is a

prime

numbers such that

m + 1

_{= 2p.}

Then

and hence

_m)/(m

+

₁₎₂

is

_p-integral.

PROOF.

_By

_(3),

Since

the last sum is

equal

to

Noting

that

we see that

because _{p -} 1 1

_(recall

that n is even and _p is

_odd).

This _proves the lemma.

PROOF OF THEOREM 3. 1. Let

_Bn

= for

n ~

1 and

Bl

=

-1/2.

Then

E"O 0

Bn Xn /n! = x/(eX - 1).

By

(2)

in the

_{introduction,}

we have

From this we see that

Since

jg"’ -

B,

= 0 for odd 1 &#x3E;

_3,

_we have for odd n

2. We make use of Theorem 1. Part

(a)

is obvious because the

_Stirling

numbers in the formula in Theorem 1 are

integers.

For the remainder of

the

_proof,

first we note that the

_expression

_{m! / (m + 1 ) 2}

in the summand of the formula is an

_integer

_except

when m + 1 =

8, 9,

_a

_prime

number,

_or 2 x

a

_{prime number,}

as can be checked in an

_elementary

_way.

_Now,

Lemma 1 tells us that _any

_prime

_{number p &#x3E;}

5

satisfying

m + 1 =

2p

does not _appear

Our next task is to consider the case m + 1 = _p, where _p is a

_prime

number &#x3E; 5. In this case

The

_righthand

side is

_congruent

modulo _p to -1 if _{p -}

_Iln

and to 0 if

p - 1

In.

Thus

if p -

11n,

the

p-order

of

m) / (m

+

1)2

is -2.

Since the other summands in the formula in Theorem 1 are

p-integral,

we

have shown

part

(b)-i.

Suppose

p - and calculate modulo

p .

Using

we see that

- , ,

It is known that

_(cf.

Cor. of

_Prop.

15.2.2 in

_[2])

if n is even and _{p - 1}

Xn,

then

On the other

_hand,

when we _{put n} mod

p - 1 = n’,1 n’ p - 1

(since

both n and _{p - 1} are _even, n’ is also

even),

we find

=

B;~1~

mod _p

_{(see (63)}

of Vandiver

_[4]

and Section 3

_below).

We therefore have

where m + 1 = _p and n’ - n mod _{p -}

1,1

n’

_{p - 1.} Since _{p - 1}

aen,

the number is

_p-integral

and mod _p

_(Prop.

15.2.4 and Th.5

_following

it in

_[2]).

Thus

r 1

This

_{readily gives}

_part

_(b)-ii

of the theorem.

The

_only

summands in Theorem 1 which may not be

_3-integral

are

2!S(n,

2)/32,

-5!S(n,

5)/62,

and

_8!S(n,

_8)/92.

_By

direct calculation

_using

the formula

_(3),

we obtain

_part

(c).

In a similar _manner, we can determine

3. A theorem of Vandiver

As an

application

of Theorems 1 and

_2,

we _prove the

following

proposi-tion

_originally

due to Vandiver.

PROPOSITION. Let p be an odd

prime

number. For 1 i _{p -}

2,

PROOF.

_By

Theorem 1 and Fermat’s little

_theorem,

we see that ¿4 B.. ,- B..

Theorem 2 says that the

_righthand

side is

_equal

to which

_by

Theorem 1 is

_equal

_2,

_rra)(rn,

+

1)2.

LEMMA 2.

_Suppose

_p is an odd

_prime,

and 1 ~ m _{p - 2.} Then

PROOF. The

_Stirling

numbers

_satisfy

the recurrence formula

Thus if we

_put

2, m)

= bm,

we

_get

But

_by

_(3),

and we thus conclude that

This

_together

with the relation

bi

=

S(p - 2,1)

= 1

gives

the lemma and hence

_completes

the

_proof

of the

_proposition.

REMARK. If i &#x3E;

_1,

the

_righthand

side of the

_proposition

is

_congruent

modulo _{p to}

and this

_being

_congruent

to

Bi

(even

when i =

1)

is _a

special

case of

Vandiver’s congruence

(63)

in

_[4].

Acknowledgement

The

_present

paper was written

_during

the author’s

_stay

in 1993 at the

university

of

_Cologne

in

_Germany,

as a research fellow of the Alexander von Humboldt foundation. He would like to thank the foundation and his host

_professor

Peter Schneider for their

_hospitality

and

_support.

REFERENCES

[1]

Gould,

H. W. :

_Explicit

formulas for Bernoulli

_numbers,

Amer. Math.

_Monthly

79

_{(1972), 44-51.}

[2]

Ireland, K. and _Rosen, M. : A Classical Introduction to Modern Number

Theory,

second edition.

_Springer

GTM 84

₍₁₉₉₀₎

[3]

Jordan,

Charles : Calculus of Finite

_Differences,

Chelsea Publ.

Co.,

New

_York,

(1950)

[4]

Vandiver,

H. S. : On

_developments

in an arithmetic

_theory

of the Bernoulli

and allied

_numbers,

_Scripta

Math. 25

_{(1961), 273-303}

Masanobu KANEKO

Graduate School of Mathematics

Kyushu University

33

Fukuoka _812-81,

_Japan