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Journal of Integer Sequences, Vol. 17 (2014),
Article 14.4.5
23 11
Multi-Poly-Bernoulli Numbers and
Finite Multiple Zeta Values
Kohtaro Imatomi, Masanobu Kaneko,1 and Erika Takeda
Graduate School of Mathematics
Kyushu University
Motooka 744, Nishi-ku
Fukuoka 819-0395
Japan
k-imatomi@math.kyushu-u.ac.jp
mkaneko@math.kyushu-u.ac.jp
eririntake@yahoo.co.jp
Abstract
We define the multi-poly-Bernoulli numbers slightly differently from the similar
numbers given in earlier papers by Bayad, Hamahata, and Masubuchi, and study their
basic properties. Our motivation for the new definition is the connection to “finite
multiple zeta values”, which have been studied by Hoffman and Zhao, among others,
and are recast in a recent work by Zagier and the second author. We write the finite
multiple zeta value in terms of our new multi-poly-Bernoulli numbers.
1
The second author is supported by the Japan Society for the Promotion of Science, Grant-in-Aid for
Scientific Research (B) 23340010.
1
1
Introduction
For any index set (k1 , . . . , kr ) with ki ∈ Z and r ≥ 1, we define two kinds of multi-poly(k ,...,k )
(k ,...,k )
Bernoulli numbers Bn 1 r and Cn 1 r by the following generating series:
Lik1 ,...,kr (1 − e−t )
1 − e−t
=
Lik1 ,...,kr (1 − e−t )
et − 1
=
∞
X
n=0
∞
X
Bn(k1 ,...,kr )
tn
,
n!
(1)
Cn(k1 ,...,kr )
tn
,
n!
(2)
n=0
where Lik1 ,...,kr (z) is the multiple polylogarithm series given by
Lik1 ,...,kr (z) =
(k)
z m1
.
m1k1 · · · mkr r
m1 >···>mr >0
X
(3)
(k)
When r = 1, the numbers Bn and Cn are “poly-Bernoulli numbers” defined and studied
in our previous work [7, 1]. When r = 1 and k1 = 1, the multiple polylogarithm is just the
(1)
(1)
usual logarithm − log(1 − z), and thus both Bn and Cn are the classical Bernoulli numbers
(1)
(1)
(with B1 = 1/2 and C1 = −1/2).
The finite multiple zeta value is the collection of truncated sums
X
mk11
p>m1 >···>mr >0
1
· · · mkr r
Q
L
modulo p for all primes p, considered in the quotient ring p Z/pZ/ p Z/pZ. (For more
details on finite multiple zeta values, we refer the reader to papers by Hoffman [6], Zhao [10],
and Kaneko-Zagier [8], among others.) In §4 we show the congruence
X
mk11
p>m1 >···>mr >0
1
(k1 −1,k2 ,...,kr )
≡ −Cp−2
k
· · · mr r
(mod p),
which reveals that each component of a finite multiple zeta value is a multi-poly-Bernoulli
number.
2
Recurrences and explicit formulas for the multi-polyBernoulli numbers
(k ,...,k )
(k ,...,k )
In this section, we first prove a simple relation between Bn 1 r and Cn 1 r and proceed
to give recursion relations and explicit formulas for the multi-poly-Bernoulli numbers. Since
2
the two generating series (1) and (2) differ only by the factor et , two obvious relations
(k ,...,k )
(k ,...,k )
between Bn 1 r and Cn 1 r are
n
n X
X
n
(k ,...,k )
(k1 ,...,kr )
(k1 ,...,kr )
n−i n
(k1 ,...,kr )
Bi 1 r .
Ci
and Cn
=
(−1)
Bn
=
i
i
i=0
i=0
(k ,...,kr )
A simpler relation expressing Bn 1
(k ,...,kr )
in terms of Cn 1
is the following.
Proposition 1. For any r ≥ 1, ki ∈ Z and n ≥ 1, we have
(k −1,k2 ,··· ,kr )
1
Bn(k1 ,...,kr ) = Cn(k1 ,...,kr ) + Cn−1
.
Proof. From the obvious identity
d
1
Lik1 ,...,kr (z) = Lik1 −1,k2 ,...,kr (z)
dz
z
(note that we allow the first index to be 0 or negative), we have
Lik1 −1,k2 ,...,kr (1 − e−t ) −t Lik1 −1,k2 ,...,kr (1 − e−t )
d
−t
Lik1 ,...,kr (1 − e ) =
·e =
,
dt
1 − e−t
et − 1
and hence
∞
X
Z tX
∞
Z
t
Lik1 −1,k2 ,...,kr (1 − e−t )
dt
n!
n!
et − 1
0 n=0
0
t
e
1
−t
−t
= Lik1 ,...,kr (1 − e ) = Lik1 ,...,kr (1 − e ) t
−
e − 1 et − 1
Lik1 ,...,kr (1 − e−t ) Lik1 ,...,kr (1 − e−t )
−
=
1 − e−t
et − 1
∞
∞
X
tn X (k1 ,...,kr ) tn
=
Bn(k1 ,...,kr ) −
.
Cn
n!
n!
n=0
n=0
(k1 −1,k2 ,...,kr ) t
Cn−1
n=1
n
=
t
Cn(k1 −1,k2 ,...,kr )
n
dt =
This proves the proposition.
Proposition 2. For any k1 , . . . , kr ∈ Z and n ≥ 0, we have
!
n−1 X
1
n
(k1 ,...,kr )
Bm
Bn(k1 ,...,kr ) =
Bn(k1 −1,k2 ,...kr ) −
m
−
1
n+1
m=1
and
Cn(k1 ,...,kr )
(−1)n
=
n+1
!
n−1
X
n
n
(k1 ,...,kr )
(k1 −1,k2 ,...,kr )
,
Cm
Cm
−
(−1)m
(−1)m
m
−
1
m
m=1
m=0
n
X
where an empty sum is understood to be 0.
3
(k ,...,k )
Proof. For Bn 1 r , we multiply the defining equation (1) by 1 − e−t and differentiate with
respect to t to obtain
∞
∞
n
n−1
X
X
e−t
−t
−t
−t
(k1 ,...,kr ) t
(k1 ,...,kr ) t
+ (1 − e )
,
Lik1 −1,k2 ,...,kr (1 − e ) = e
Bn
Bn
1 − e−t
n!
(n − 1)!
n=0
n=1
and from this we have
∞
X
t
Bn(k1 ,...,kr )
n
n!
n=0
=
∞
X
t
Bn(k1 −1,k2 ,...,kr )
n
n!
n=0
t
− (e − 1)
∞
X
Bn(k1 ,...,kr )
n=1
tn−1
.
(n − 1)!
(k ,...,kr )
Comparing the coefficients of tn /n! on both sides, we obtain the desired relation for Bn 1
(k ,...,k )
The relation for Cn 1 r is obtained similarly.
.
We make a remark on the following point: since we allow k1 to be positive or negative,
the recursions above need never ‘stop’. However, by the identity
Li0,k2 ,...,kr (z) =
(0,k ,...,k )
z
Lik2 ,...,kr (z),
1−z
(0,k ,...,k )
(k ,...,k )
the Bn 2 r or Cn 2 r can be written as simple linear combinations of Bm 2 r or
(k ,...,k )
(k ,...,k )
Cm 2 r (0 ≤ m < n). The proposition therefore gives a way to express Bn 1 r or
(k ,...,k )
Cn 1 r in terms of numbers with smaller n and lower “depth” (length r of the index set).
Next, we write our multi-poly-Bernoulli numbers as finite sums involving
h n iStirling numbers
and the second
of the second kind. Recall that the Stirling numbers of the first kind
m
n
are the integers uniquely determined by the following recursions for all integers
kind
m
m, n (see Knuth [9]):
0
n
0
= 1,
=
= 0 (n, m 6= 0),
0
0
m
0
n
0
= 0, (n, m 6= 0),
=
= 1,
m
0
0
n+1
n
n
=
+n
(∀n, m ∈ Z),
m
m−1
m
n
n
n+1
(∀n, m ∈ Z).
+m
=
m
m−1
m
Theorem 3. For any r ≥ 1, k1 , k2 , · · · , kr ∈ Z, and n ≥ 0, we have
Bn(k1 ,...,kr )
= (−1)
n
(−1)m1 −1 (m1 − 1)!
X
(−1)m1 −1 (m1 − 1)!
n+1≥m1 >m2 >···>mr >0
and
Cn(k1 ,...,kr ) = (−1)n
X
n+1≥m1 >m2 >···>mr >0
4
n
m1 −1
m 1 k1 · · · m r kr
mk11
· · · mkr r
n+1
m1
.
Proof. Using the well-known generating series (cf. Graham et al. [3, §7.4])
∞ n
X
n t
(et − 1)m
=
(m ≥ 0),
m!
m n!
n=m
we have
∞
X
Bn(k1 ,...,kr )
n=0
tn
Lik1 ,··· ,kr (1 − e−t )
=
n!
1 − e−t
X
(1 − e−t )m1 −1
=
m 1 k1 · · · m r kr
m >m >···>m >0
1
r
2
∞
(−1)m1 −1 (m1 − 1)! X
n
(−t)n
=
m 1 k1 · · · m r kr
n!
m1 − 1
m1 >m2 >···>mr >0
n=m1 −1
∞
X
X
(−1)m1 −1 (m1 − 1)! m1n−1 tn
n
=
(−1)
.
m 1 k1 · · · m r kr
n!
n=r−1
n+1≥m >m >···>m >0
X
1
2
r
(k ,...,k )
Comparing the coefficients of tn , we obtain the formula for Bn 1 r .
(k ,...,k )
To obtain the formula for Cn 1 r , we proceed similarly by using the equation
∞
X
n + 1 tn
et (et − 1)m−1
=
(m ≥ 1),
(m − 1)!
n!
m
n=m−1
which follows from equation (4) by differentiation:
∞
X
n=0
Cn(k1 ,...,kr )
Lik1 ,...,kr (1 − e−t )
tn
=
n!
et − 1
X
e−t (1 − e−t )m1 −1
=
m 1 k1 · · · m r kr
m >m >···>m >0
1
2
r
∞
(−1)m1 −1 (m1 − 1)! X
n + 1 (−t)n
=
m 1 k1 · · · m r kr
n!
m1
m1 >m2 >···>mr >0
n=m1 −1
∞
X
X
(−1)m1 −1 (m1 − 1)! n+1
tn
m1
n
=
(−1)
.
m 1 k1 · · · m r kr
n!
n=r−1
n+1≥m >m >···>m >0
X
1
2
r
We end this section by giving formulas for the special case k1 = · · · = kr = 1.
Proposition 4. For any r ≥ 1 and n ≥ r − 1, we have
r
r
z }| {
z }| {
n
+
1
n+1
1
1
(1)
(1)
(1,1,...,1)
(1,1,...,1)
Bn
Bn−r+1 , Cn
Cn−r+1 .
=
=
n+1
n+1
r
r
5
(4)
Proof. Use the identity Li 1,...,1 (z) = (− log(1 − z))r /r! to obtain
|{z}
r
r
∞
X
∞
z}|{ n
tr−1
tr−1 X (1) tn
t
( 1,...,1) t
Bn
B
=
=
n!
r! 1 − e−t
r! n=0 n n!
n=0
and
∞
X
n=0
The proposition follows.
3
r
∞
tr−1 t
tr−1 X (1) tn
C
=
=
.
n!
r! et − 1
r! n=0 n n!
t
Cn( 1,...,1)
z}|{
n
Multi-poly-Bernoulli numbers with negative indices
Proposition 5. 2 We have the following generating series identities for multi-poly-Bernoulli
numbers with non-positive upper indices:
1)
∞
X
∞
X
Bn(−k1 ,...,−kr )
k1 ,...,kr =0 n=0
=
(e−x1
tn xk11
x kr
··· r
n! k1 !
kr !
(1 − e−t )r−1
,
+ e−t − 1)(e−x1 −x2 + e−t − 1) · · · (e−x1 −···−xr + e−t − 1)
2)
∞
X
∞
X
k1 ,...,kr =0 n=0
=
(e−x1
x kr
xk11
··· r
n! k1 !
kr !
t
Cn(−k1 ,...,−kr )
n
e−t (1 − e−t )r−1
.
+ e−t − 1)(e−x1 −x2 + e−t − 1) · · · (e−x1 −···−xr + e−t − 1)
Proof. Use the defining identity
∞
X
n=0
2
Bn(−k1 ,...,−kr )
X
tn
= (1 − e−t )−1
n!
m >···>m
1
mk11 · · · mkr r (1 − e−t )m1
r >0
This proposition and the corollary below are also pointed out by Genki Shibukawa.
6
to obtain
∞
X
∞
X
X
xkr r
xk11
···
= (1 − e−t )−1
n! k1 !
kr !
m >···>m
t
Bn(−k1 ,...,−kr )
k1 ,...,kr =0 n=0
= (1 − e−t )−1
= (1 − e−t )−1
n
1
∞
X
(1 − e−t )m1 em1 x1 +···+mr xr
r >0
(1 − e−t )n1 +···+nr e(n1 +···+nr )x1 +(n2 +···+nr )x2 +···+nr xr
n1 ,...,nr =1
∞
X
(1 − e−t )ex1
n1 ,...,nr =1
n1
(1 − e−t )ex1 +x2
n2
· · · (1 − e−t )ex1 +···+xr
nr
ex1 +···+xr
ex1 +x2
e x1
···
1 − (1 − e−t )ex1 1 − (1 − e−t )ex1 +x2
1 − (1 − e−t )ex1 +···+xr
(1 − e−t )r−1
.
+ e−t − 1)(e−x1 −x2 + e−t − 1) · · · (e−x1 −···−xr + e−t − 1)
= (1 − e−t )r−1
=
(e−x1
(−k ,...,−k )
r
The identity for Cn 1
readily follows from this because the defining generating series
differ only by the factor e−t .
Corollary 6. For any integers k1 , . . . , kr ≥ 0 and n ≥ 0, the multi-poly-Bernoulli numbers
(−k ,...,−kr )
(−k ,...,−kr )
Bn 1
and Cn 1
are positive integers.
Proof. The right-hand sides of 1) and 2) of the proposition can be rewritten as
t
t
e (e − 1)
r−1
ex1 +x2
ex1 +···+xr
e x1
·
···
·
1 − (et − 1)(ex1 − 1) 1 − (et − 1)(ex1 +x2 − 1)
1 − (et − 1)(ex1 +···+xr − 1)
and
(et − 1)r−1 ·
ex1 +x2
ex1 +···+xr
e x1
·
·
·
·
1 − (et − 1)(ex1 − 1) 1 − (et − 1)(ex1 +x2 − 1)
1 − (et − 1)(ex1 +···+xr − 1)
(−k ,...,−k )
(−k1 ,...,−kr )
r
respectively, from which the positivity of Bn 1
and Cn
are integers follows from the explicit formulas in Theorem 3.
is obvious. That both
We can give some explicit formulas different from Theorem 3 for special indices.
Proposition 7. For r ≥ 1, k ≥ 0 and n ≥ 0, we have
1)
r−1
n
X
n
,
j
n+1
.
j+1
z}|{
j
(−1) (j + 1) j!
= (−1)
r−1
j=0
z}|{
j
= (−1)
(−1) (j + 1) j!
r−1
j=0
Bn(−k, 0,...,0)
n
r−1
Cn(−k, 0,...,0)
n
j
n
X
j
7
k
k
2)
r−1
min{n+1−r,k}
z}|{
Bn( 0,...,0,−k)
=
z}|{
X
n
(r + j − 1)!j!
r+j−1
min{n+1−r,k}
Cn( 0,...,0,−k)
=
k+1
,
j+1
n+1
(r + j − 1)!j!
r+j
j=0
r−1
X
j=0
k+1
.
j+1
Proof. Set x2 = · · · = xr = 0 in 1) of Proposition 5 to obtain
∞ X
∞
X
k=0 n=0
r−1
(1 − e−t )r−1
xk
= −x
.
n! k!
(e + e−t − 1)r
t
Bn(−k, 0,...,0)
z}|{
n
From this we have
∞ X
∞
X
r−1
r−1
∞ z}|{ n k
X
1
1 − e−t
r−1
(−k, 0,...,0) t x
y
=
y r−1
Bn
−x + e−t − 1)
−x + e−t − 1
n!
k!
(e
e
r=1 k,n=0
r=1
1
1
−t
−t
1−e
+ e − 1) 1 − e−x +e−t
y
−1
1
= −x
−t
e + e − 1 − (1 − e−t )y
ex
=
1 − ex (1 + y)(1 − e−t )
∞
X
=
e(j+1)x (1 + y)j (1 − e−t )j
=
(e−x
j=0
∞
X
∞
X
n
n t
=
e
(1 + y) (−1) j!
(−1)
j n!
j=0
n=j
n
∞ X
n
X
n t
(j+1)x
j
n+j
=
e
(1 + y) (−1) j!
.
j
n!
n=0 j=0
Comparing the coefficients of
tn xk r−1
y
n! k!
(j+1)x
j
j
n
(−k,0,...,0)
on both sides, we obtain 1) for Bn
8
.
Similarly, we compute
∞ X
∞
X
r−1
z}|{
ex
xk r−1
y
= e−t ·
n! k!
1 − ex (1 + y)(1 − e−t )
t
Cn(−k, 0,...,0)
r=1 k,n=0
n
=
∞
X
j=0
∞
X
e(j+1)x (1 + y)j e−t (1 − e−t )j
∞
X
n + 1 tn
=
e
(1 + y) (−1) j!
(−1)
j + 1 n!
j=0
n=j
∞ X
n
X
n + 1 tn
(j+1)x
j
n+j
=
e
(1 + y) (−1) j!
j + 1 n!
n=0 j=0
(j+1)x
j
j
n
(−k,0,...,0)
and obtain the formula for Cn
.
For 2), we set x1 = · · · = xr−1 = 0 in the formulas in Proposition 5 and have
∞ X
∞
X
k=0 n=0
r−1
z}|{
Bn( 0,...,0,−k)
(et − 1)r−1
t n xk
= −x
n! k!
e + e−t − 1
= (et − 1)r−1 ·
=
∞
X
et+x
1 − (et − 1)(ex − 1)
et (et − 1)r+j−1 ex (ex − 1)j
j=0
∞
∞
∞ X
X
X
n + 1 tn
k + 1 xk
=
(r + j − 1)!
· j!
r
+
j
j + 1 k!
n!
j=0
n=r+j−1
k=j
and
∞
∞ X
X
k=0 n=0
r−1
z}|{
n
∞
∞ ∞
X
X
X
k + 1 xk
xk
n
t
=
· j!
.
(r + j − 1)!
j
+
1
n! k!
n!
k!
r
+
j
−
1
j=0
n=r+j−1
k=j
t
Cn( 0,...,0,−k)
n
Comparing the coefficients of
4
t n xk
n! k!
on both sides, we obtain 2).
Connection to the finite multiple zeta values
As recalled in Q
the introduction,
the finite multiple zeta value ζA (k1 ,. . . , Q
kr ) is an element in
L
the ring A := p Z/pZ/ p Z/pZ represented by ζA (k1 , . . . , kr )(p) p ∈ p Z/pZ where
ζA (k1 , . . . , kr )(p) :=
X
mk11
p>m1 >···>mr >0
9
1
mod p.
· · · mkr r
Theorem 8. For any r ≥ 1 and ki ∈ Z, we have
(k −1,k2 ,...,kr )
1
ζA (k1 , . . . , kr )(p) = −Cp−2
mod p.
More generally, we have for any r ≥ 1, j ≥ 0, and ki ∈ Z
(k −1,k2 ,...,kr )
1
ζA (1, . . . , 1, k1 , . . . , kr )(p) = −Cp−j−2
| {z }
mod p.
j
Proof. By the explicit formula in Theorem 3, we have
(k1 −1,k2 ,...,kr )
Cp−2
(−1)m1 −1 (m1 − 1)!
X
= −
m1k1 −1 mk22
p−1≥m1 >m2 >···>mr >0
=
(−1)m1 m1 !
X
mk11 mk22
p−1≥m1 >m2 >···>mr >0
m1
k
r
· · · mr
p−1
m1
· · · mkr r
p−1
.
By the well-known formula (cf. Graham et al. [3, §6.1])
X
m1
p−1
l m1
m1
lp−1 ,
(−1)
=
(−1) m1 !
l
m1
l=0
we see
(−1)
m1
p−1
m1 !
m1
m1
X
m1
(−1)
≡
l
l=1
l
(mod p),
the sum on the right being equal to (1 − 1)m1 − 1 = −1. This proves that
(k −1,k2 ,...,kr )
1
Cp−2
mod p = −ζA (k1 , . . . , kr )(p) .
For the second identity, we proceed as follows. First note
X
1
i · · · ij mk11 · · · mkr r
p−1≥i1 >···>ij >m1 >···>mr ≥1 1
X
X
1
1
=
.
k1
kr
i
·
·
·
i
m
·
·
·
m
1
j
1
r
p−1≥i1 >···>ij >m1
p−j>m1 >···>mr ≥1
By changing il → p − il , we see that
X
X
1
≡ (−1)j
i · · · ij
p−m >i >···>i
p−1≥i >···>i >m 1
1
j
1
1
j
1
1
i · · · ij
≥1 1
(mod p).
Using the formula
X
p−m1 >ij >···>i1
1
p − m1
1
,
=
j
+
1
i
·
·
·
i
(p
−
m
−
1)!
1
j
1
≥1
10
and the congruences
n
p−m
≡
(mod p) (1 ≤ m ≤ n ≤ p − 1)
m
p−n
(see Hoffman [6, §5]) and
1
≡ (−1)m1 +1 m1 !
(p − m1 − 1)!
(mod p),
we obtain (for odd p)
ζA (1, . . . , 1, k1 , . . . , kr )(p) =
| {z }
j
X
(−1)j+m1 +1 m1 !
mk11 · · · mkr r
p−j>m1 >···>mr ≥1
(k −1,k2 ,...,kr )
1
= −Cp−j−2
p−j−1
m1
mod p
mod p.
This concludes the proof of the theorem.
Combining the theorem with Proposition 4, we see that
k−1
k−2
z }| {
ζA (2, 1, . . . , 1)
=
=
z}|{
( 1,...,1)
−Cp−2
mod p
(Bp−k mod p)p .
p
p−1
1
Bp−k mod p
= −
p−1 k−1
p
p−1
(We used k−1
≡ (−1)k−1 (mod p) and (−1)p−k Bp−k = Bp−k for large enough p.) As is
discussed in Kaneko-Zagier [8], the element (Bp−k mod p)p on the right is regarded as an
analogue of kζ(k). (Note that the obvious analogue ζA (k) of ζ(k) is 0.)
References
[1] T. Arakawa and M. Kaneko, Multiple zeta values, poly-Bernoulli numbers, and related
zeta functions, Nagoya Math. J. 153 (1999), 189–209.
[2] A. Bayad and Y. Hamahata, Multiple polylogarithms and multi-poly-Bernoulli polynomials, Funct. Approx. Comment. Math. 46 (2012), 45–61.
[3] R. Graham, D. Knuth, and O. Patashnik, Concrete Mathematics, Addison-Wesley, 1989.
[4] Y. Hamahata and H. Masubuchi, Special multi-poly-Bernoulli numbers, J. Integer Seq.
10 (2007), Article 07.4.1.
[5] Y. Hamahata and H. Masubuchi, Recurrence formulae for multi-poly-Bernoulli numbers,
Integers 7 (2007), #A46.
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[6] M. Hoffman, Quasi-symmetric functions and mod p multiple harmonic sums, preprint,
http://arxiv.org/abs/math/0401319.
[7] M. Kaneko, Poly-Bernoulli numbers, J. Théor. Nombres Bordeaux 9 (1997), 199–206.
[8] M. Kaneko and D. Zagier, Finite multiple zeta values, in preparation.
[9] D. Knuth, Two notes on notation, Amer. Math. Monthly 99 (1992), 403–422.
[10] J. Zhao, Wolstenholme type theorem for multiple harmonic sums, Int. J. Number Theory
4 (2008), 73–106.
2010 Mathematics Subject Classification: Primary 11B68; Secondary 11M32.
Keywords: multi-poly-Bernoulli number, Stirling number, finite multiple zeta value.
(Concerned with sequences A027643, A027644, A027645, A027646, A027647, A027648, A027649,
A027650, and A027651.)
Received October 24 2013; revised version received February 17 2014. Published in Journal
of Integer Sequences, February 17 2014.
Return to Journal of Integer Sequences home page.
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