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CLOSED FORMULA FOR POLY-BERNOULLI NUMBERS
Roberto Sanchez-Peregrino
Dipartimento di Matematica Pura ed Applicata, Via Belzoni 7,1-35131 Padova, Italy
e-mail: [email protected]
(Submitted August 2000-Final Revision April 2001)
1. INTRODUCTION AND BACKGROUND.
In the present note we shall give two proofs of a property of the poly-Bernoulli numbers, the
closed formula for negative index poly-Bernoulli numbers given by Arakawa and Kaneko [1].
The first proof uses weighted Stirling numbers of the second kind (see [2], [3]). The second,
much simpler, proof is due to Zeilberger.
In Kaneko's paper, "On Poly-Bemoulli Numbers" [5], the poly-Bernoulli numbers, which
generalize the classical Bernoulli numbers, are defined and studied. For every integer k9 called the
index, we define a sequence of rational numbers B^ (« = 0,1,2,...), which we refer to as polyBernoulli numbers, by
l L i fc (.)| z=1 _ e ., = | B ^ .
(1)
Here, for any integer k9 Llk(z) denotes the formal power series ZJ^z'V/w*, which is the k^
polylogarithm if k > 1 and a rational function if k < 0. When k =1, B^ is the usual Bernoulli
number (with B} = 1 /2). In [4] Kaneko obtained an explicit formula for Bkn:
where {%} is an integer referred to as a Stirling number of the second kind [6].
2, CLOSED FORMULA
Theorem 2d (Closed Formula): For any w, k > 0, we have
**-iw{j:!}C:i}-
p>
We need two lemmas. We use the notation and numeration of the equations in Carlitz's
paper [3].
Lemma 2.1:
ii-Wrifyfy = H)"'!{m} = H)"^R(», t, i),
(4)
where
Proof: In order to prove this lemma, we calculate the generating function:
i0£<-H"){s}S-s<-<7K{s}S
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CLOSED FORMULA FOR POLY-BERNOULLI NUMBERS
:
Z, ( _1 )^ I I P1-
=
ji
/ i _ / i „ z\y+i> ^y * e g ener alized binomial theorem,
= e-*(e-* - if = £*!R(/i, *, lX-l)"»7 ? by [3], (3.9).
Lemma 2.2:
£ZBw t *y = Z/»y(*)^0')
(5)
where p / x ) = 7! I " 0 R(n, j , l)x".
Proof: By (2), we have
Z Z B - V / = £ 1 1 ; f n r H r ^ W i ) * W , by pi, (3.4),
00
00
CO
00
/
\
/
R
n
f
1 \
= Z Z Z E"(7 (M,o (-D (-i)^'CW
w =0
&=Q m=0 *=0
V /
•
V
I
JJ
= Z | ) A 0 ' X - l ) " | ; f 7 l ( - i r ™ l f c } « " , by Lemma 2.1,
= Z ZftCy)(-ir(-i)"^!R(», Al)*" = X ACO'ifX",', i)*" = fipt(x)pt(y).
£=0 n=Q
£=0
n=0
£=0
Proof of (3): To prove (3), we compare the coefficients on both sides of (5). In the course
of Arakawa and Kaneko's proof they prove the following proposition.
Proposition 2.1: For n > 05
I(-l)'Bft = 0.
^=0
Proof: We offer a more direct proof:
£=0
£=0
m=0
^
J
^ - i r l i l - i r ^ ^ + l / h l , by [4], (6.20),
= (-Dn ±{-\)mm\ {;+Jl=nys l n + l =o.
3. ANOTHER PROOF
In Kaneko's paper [4], he obtained the symmetric formula:
Y Ywk
2002]
— *—= -
(6)
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CLOSED FORMULA FOR POLY-BERNOULLI NUMBERS
By using (6), D. Zeilberger gives a much simpler proof of (3) as follows:
-k x y
fc>Ow>o n. K\
Qx+y
e -te
e
y>0
= S T T T ^ O + W - eXy<rex)ij+1)0 - ^ y (-v)
= I T-^Dja-o^D.ta-^)^ 1 ].
/so U + V
Now using the usual generating function for the Stirling numbers of the second kind {£}, i.e.,
sr [n\un
_{eu-lf
n>k
he obtains:
jn
,Jfc
(-iy+ic/+i)i ^ , U + lf «i
«>0fc>0
xD,
(-iy+1o+D! z { / + I } F
-z^zfriftzfriHi
ACKNOWLEDGMENT
The author expresses his gratitude to D. Zeilberger for advising and permitting him to include
the proof in this paper and he is very grateful to the anonymous referee for useful comments.
REFERENCES
1.
2.
3.
4.
5.
6.
T. Arakawa & M. Kaneko. "On Poly-Bemoulli Numbers.11 Comment Math. Univ. St. Paul
48.2 (1999): 159-67.
A. Z. Broder. "The r»Stirling Numbers." Discrete Math. 49 (1984):241-59.
L. Carlitz. "Weighted Stirling Numbers of the First and Second Kind-I." The Fibonacci
Quarterly 18.2 (1980): 147-62.
R. L. Graham, D. E. Knuth, & O. Patashnik. Concrete Mathematics. 2nd ed. Reading, MA:
Addison Wesley, 1989.
M. Kaneko. "Poly-Bernoulli Numbers." Journal de Theorie des Nomhres de Bordeaux 9
(1997):221-28.
D. Knuth. "Two Notes on Notation." Amer. Math. Monthly 99 (1992):403-22.
AMS Classification Numbers: 11A07, 11B73
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