ON FUNCTIONS OF ARAKAWA AND KANEKO AND MULTIPLE ZETA
FUNCTIONS
MARKUS KUBA
R ∞ ts−1
1
−t )dt,
A BSTRACT. We study for s ∈ N the functions ξk (s) = Γ(s)
0 et −1 Lik (1−e
R
s−1
∞ t
1
−t )dt, introduced
and more generally ξk1 ,...,kr (s) = Γ(s)
0 et −1 Lik1 ,...,kr (1 − e
by Arakawa and Kaneko [2] and relate them with (finite) multiple zeta functions, partially answering a question of [2]. In particular, we give an alternative proof of a result of
Ohno [8].
1. I NTRODUCTION
Let Lik1 ,...,kr (z) denote the multiple polylogarithm function defined by
X
z n1
,
Lik1 ,...,kr (z) =
k1 k2
n n2 . . . nkr r
n1 >n2 >···>nr ≥1 1
with k1 ∈ N \ {1} and ki ∈ N = {1, 2, . . . }, 2 ≤ i ≤ r, and |z| ≤ 1. For z = 1 the
multiple polylogarithm function Lik1 ,...,kr (1) = ζ(k1 , . . . , kr ) simplifies to a multiple zeta
function, also called multiple zeta value, where ζ(k1 , . . . , kr ) and ζN (k1 , . . . , kr ) denote
the (finite) multiple zeta function defined by
X
1
ζ(k1 , . . . , kr ) =
,
k1 k2
kr
n
n
2 . . . nr
n1 >n2 >···>nr ≥1 1
X
1
ζN (k1 , . . . , kr ) =
,
ka1 k2
n n2 . . . nkr r
N ≥n1 >n2 >···>n` ≥1 1
with k1 ∈ N \ {1}, and k2 , . . . , kr ∈ N for the infinite series and N, k1 , . . . , kr ∈ N for the
finite counterpart. Arakawa and Kaneko [2] introduced the function ξk (s), and the more
general function ξk1 ,...,kr (s), defined by
Z ∞ s−1
1
t
Lik (1 − e−t )dt,
ξk (s) =
Γ(s) 0 et − 1
Z ∞ s−1
1
t
ξk1 ,...,kr (s) =
Lik1 ,...,kr (1 − e−t )dt,
Γ(s) 0 et − 1
respectively, being absolut convergent for <(s) > 0, and related them for special choices
of s and k1 , . . . kr to multiple zeta functions. Ohno [8] obtained a result for ξk (n), with
n ∈ N, using his generalization of the duality and sum formulas for multiple zeta functions.
We will provide for n ∈ N evaluations of the function ξk1 ,...,kr (n) to multiple zeta
functions, partially answering a question of [2]; in particular we give a short and simple
proof of Ohno’s result [8]. For the evaluation of the general case ξk1 ,...,kr (n) we use a
finite version of the so-called stuffle identity for multiple zeta functions. Subsequently,
we will utilise a variant of (finite) multiple zeta functions, called the multiple zeta star
∗
functions or non-strict multiple zeta functions ζN
(k1 , . . . , kr ), which recently attracted
2000 Mathematics Subject Classification. 11M06, 40B05.
Key words and phrases. Multiple zeta functions, finite multiple zeta functions, multiple zeta star functions,
finite stuffle identity.
This work was supported by the Austrian Science Foundation FWF, grant S9608-N13.
1
2
M. KUBA
some interest,[1, 8, 10, 9, 6, 7, 5, 11] where the summation indices satisfy N ≥ n1 ≥
n2 ≥ · · · ≥ nr ≥ 1 in contrast to N ≥ n1 > n2 > · · · > nr > 1, as in the usual
definition (1),
X
1
∗
ζN
,
(k1 , . . . , kr ) =
k1 k2
n n2 . . . nkr r
N ≥n1 ≥n2 ≥···≥nr ≥1 1
with N, k1 , . . . , kr ∈ N. The star form can be converted into ordinary finite multiple zeta
functions by considering all possible deletions of commas, i.e.
X
`1
`2
r
r
X
X
X
X
∗
ζN (k1 , . . . , kr ) =
k i1 ,
k i2 , . . . ,
ζN
kih ;
i1 =1
h=1 1≤`1 <`2 <···<`h−1 <r
i2 =`1 +1
ih =`h−1 +1
(1)
note that the first term h = 1 should be interpreted as ζN ( i1 =`0 +1 ki1 ), subject to `0 = 0.
∗
The notation ζN
(k1 , . . . , kr ) is chosen in analogy with Aoki and Ohno [1] where infinite
∗
counterparts of ζN
(k1 , . . . , kr ) have been treated. First we will study the instructive case
of ξk (n), reproving the result of Ohno. Then we will state our main result concerning the
evaluation of ξk1 ,...,kr (n) into multiple zeta functions.
Pr
2. A SIMPLE EVALUATION OF ξk (n)
Ohno [8] evaluated the sum ξk (n) for k, n ∈ N applying his generalization of the duality
and sum formulas for multiple zeta functions to a result of Arakawa and Kaneko [2]. In
the following we will give an alternative simple and self-contained derivation of his result,
stated in Theorem 1. In order to evaluate ξk (n) for k, n ∈ N we only use the two basic
facts stated below.
Z ∞
1
1
tn−1 e−t` dt = n , for `, n ∈ N,
Γ(n) 0
`
n (2)
`−1
X
n (−1)
∗
∗
=
ζ
(1,
.
.
.
,
1
)
=
ζ
({1}
)
for
r
∈
N.
r
n
n
| {z }
`
`r
`=1
r
The second identity can be immediately deduced by repeated usage of the formula nk =
Pn
`−1
`=k k−1 . We proceed as follows.
Z ∞ n−1
Z ∞
X (1 − e−t )m−1
1
1
t
−t
n−1 −t
ξk (n) =
Li
(1
−
e
)dt
=
t
e
dt.
k
Γ(n) 0 et − 1
Γ(n) 0
mk
m≥1
We expand (1 − e−t )m−1 by the binomial theorem and interchange summation and integration. According to (2) we obtain
X 1 m−1
X m − 1 (−1)` Z ∞
tn−1 e−(`+1)t dt
ξk (n) =
mk
`
Γ(n) 0
m≥1
`=0
X 1 m−1
X m − 1 (−1)`
=
.
mk
`
(` + 1)n
m≥1
Since
m−1
`
=
m `+1
`+1 m ,
`=0
we get according to (2) after an index shift the following result
Theorem 1 (Ohno [8]). For k, n ∈ N the function ξk (n) is given by
X
1
ξk (n) =
= ζ ∗ (k + 1, {1}n−1 ).
k+1
m
m
.
.
.
m
2
n
1
m1 ≥m2 ≥···≥mn ≥1
ON FUNCTIONS OF ARAKAWA AND KANEKO AND MULTIPLE ZETA FUNCTIONS
3
Note that one can convert the multiple star zeta function above into ordinary multiple
zeta functions according to (1) (with respect to the corresponding relation for infinite series), or can directly simplify the multiple zeta star function using (cycle) sum formulas,
see i.e. Ohno and Wakabayashi [10] or Ohno and Okuda [9].
3. G ENERAL CASE
In the general case of ξk1 ,...,kr (n) we will prove the following result.
Theorem 2. For k1 , . . . , kr , n ∈ N the function ξk1 ,...,kr (n) is given by
ξk1 ,...,kr (n) =
X ζn∗ ({1}n−1 )ζn1 −1 (k2 , . . . , kr )
1
nk11 +1
n1 ≥1
.
Furthermore, ξk1 ,...,kr (n) can be evaluated into sums of multiple zeta functions.
The explicit evaluation of ξk1 ,...,kr (n) will be given in Corollary 1. We have
Z ∞ n−1
1
t
Lik1 ,...,kr (1 − e−t )dt
ξk1 ,...,kr (n) =
Γ(n) 0 et − 1
Z ∞
X
1
(1 − e−t )n1 −1
=
tn−1 e−t
dt.
Γ(n) 0
nk1 nk22 . . . nkr r
n1 >n2 >···>nr ≥1 1
Proceeding as before we expand (1 − e−t )n1 −1 by the binomial theorem and interchange
summation and integration. We get
Z
nX
1 −1 1
n1 − 1 (−1)` ∞ n−1 −(`+1)t
t
e
dt.
ξk1 ,...,kr (n) =
Γ(n) 0
`
nk1 nk22 . . . nkr r `=0
n1 >n2 >···>nr ≥1 1
X
According to (2) we obtain
X
ζn∗1 ({1}n−1 )
n1 >n2 >···>nr ≥1
nk11 +1 nk22 . . . nkr r
ξk1 ,...,kr (n) =
X ζn∗ ({1}n−1 )ζn1 −1 (k2 , . . . , kr )
1
=
n1 ≥1
nk11 +1
.
This proves the first part of Theorem 2.
Concerning the second part we proceed as follows. We will evaluate the product S of
finite multiple zeta (star) functions
S = Sn1 (n, k2 , . . . , k2 ) = ζn∗1 ({1}n−1 )ζn1 −1 (k2 , . . . , kr )
into sums of finite multiple zeta functions ζn1 −1 (f ), for some f = (f1 , . . . , fj ), with
fi ∈ N, 1 ≤ i ≤ j, which will prove the second part of the stated result. By (1) we can
write ζn∗1 ({1}n−1 ) in terms of ordinary finite multiple zeta functions
ζn∗1 ({1}n−1 )
=
n−1
X
X
ζn1 `1 , `2 − `1 , . . . , n − `h−1 − 1 ;
h=1 1≤`1 <`2 <···<`h−1 <n−1
for example ζn∗1 ({1}3 ) = ζn1 (3) + ζn1 (1, 2) + ζn1 (2, 1) + ζn1 (1, 1, 1). We can convert
finite zeta functions ζN (a1 , . . . , ar ) into finite zeta functions ζN −1 (b1 , . . . , bs ) by
ζN (a1 , . . . , ar ) = ζN −1 (a1 , . . . , ar ) +
1
ζN −1 (a2 , . . . , ar ).
N a1
4
M. KUBA
Consequently, we can express the product S = ζn∗1 ({1}n−1 )ζn1 −1 (k2 , . . . , kr ) of finite
multiple zeta (star) functions in the following way.
S=
n−1
X
X
ζn1 −1 `1 , `2 − `1 , . . . , n − `h−1 − 1 ζn1 −1 (k2 , . . . , kr )
h=1 1≤`1 <`2 <···<`h−1 <n−1
+
n−1
X
X
ζn1 −1 `2 , . . . , n − `h−1 − 1 ζn1 −1 (k2 , . . . , kr ) n`11
h=1 1≤`1 <`2 <···<`h−1 <n−1
.
(3)
Now we use finite versions of so-called stuffle identities, see i.e. Borwein et al. [3]. Stuffle
identities provide evaluation of productsPof multiple zeta functions ζ(k)ζ(h) into into sums
of multiple zeta functions ζ(k)ζ(h) = f ∈stuffle(k,h) ζ(f ); here ζ(k) = ζ(k1 , . . . , kr ) and
ζ(h) = ζ(h1 , . . . , hs ). For our problem we need finite versions of the stuffle identities,
providing evaluations of products of finite multiple zeta
Pfunctions ζN (k)ζN (h) into into
sums of finite multiple zeta functions ζN (k)ζN (h) = f ∈stuffle(k,h) ζN (f ). This would
lead then to an evaluation of ξk1 ,...,kr (n) into single finite multiple zeta functions.
Following [3] we define for two given strings k = (k1 , . . . , kr ) and h = (h1 , . . . , hs )
the set stuffle(k, h) as the smallest set of strings over the alphabet A, defined by
A = {k1 , . . . , kr , h1 , . . . , hs , “+00 , “,00 , “(00 , “)00 }
satisfying (k, h) = (k1 , . . . , kr , h1 , . . . , hs ) ∈ stuffle(k, h) and further if a string of the
form (U, kn , hm , V ) ∈ stuffle(k, h), then so are the strings (U, hm , kn , V ) ∈ stuffle(k, h)
and (U, kn + hm , V ) ∈ stuffle(k, h). Stuffle identities arise from the definition of (finite)
multiple zeta functions in terms of sums; the term stuffle derives from the manner in which
the two upper strings are combined. Other closely related identities are due to different
representations of multiple zeta functions (see for example [3]). We will use the following
result
Lemma 1 (Stuffle identity; finite version). Let ζN (k) = ζN (k1 , . . . , kr ) and ζN (h) =
ζN (h1 , . . . , hs ), with N, r, s ∈ N and ki , hj ∈ N, 1 ≤ i ≤ r, 1 ≤ j ≤ s. Then,
X
ζN (k)ζN (h) =
ζN (f ).
(4)
f ∈stuffle(k,h)
Remark 1. Note that the conditions on the parameters ki , hj can be relaxed in various
ways, i.e. ki , hj ∈ R+ . Typical examples of (finite) stuffle identities read as follows,
ζN (r)ζN (t) = ζN (r, t) + ζN (t + r) + ζN (t, r),
ζN (r, s)ζN (t) = ζN (r, s, t) + ζN (r, s + t) + ζN (r, t, s) + ζN (r + t, s) + ζN (t, r, s);
we refer to [3] concerning infinite counterparts, and also to Borwein and Girgensohn [4]
where the second identity is implicitly derived. Stuffle identities for finite multiple zeta
functions seem to be natural, since one does not have to exclude the cases k1 = 1 or
h1 = 1 in contrast to infinite multiple zeta functions.
Proof of Lemma 1 (Sketch). In order prove this result in an elementary way, one can use
induction with respect to the total length |k| + |h|. We do not want to give a full proof,
since we believe that (4) is already known (although we did not find a suitable reference in
the literature for the finite version of the stuffle identity); hence, we only sketch the simple
arguments. By definition
nr−1 −1
n1 −1
n
X
X 1
1 X
1
1
=
·
·
·
.
ζN (k) =
k1
k2
kr
nk1 nk22 . . . nkr r
nr =1 nr
n1 =1 n1 n2 =1 n2
N ≥n1 >n2 >···>nr ≥1 1
X
ON FUNCTIONS OF ARAKAWA AND KANEKO AND MULTIPLE ZETA FUNCTIONS
5
Consequently,
ζN (k)ζN (h) =
nr−1 −1
n1 −1
N
n
X 1 X
X
1
1
1 X
·
·
·
ζ
(h , . . . , hs ).
k1
k2
kr
h1 m1 −1 2
nr =1 nr m1 =1 m1
n1 =1 n1 n2 =1 n2
Since by definition of ζN (k) the summation ranges are given by N ≥ n1 > n2 > · · · >
nr ≥ 1, we can simple split the summation range of m1 into the following parts
nr−` −1
nX
N
r−1
N
r −1
X
X
X
X
X X
=
+
+
+
+
.
m1 =1
m1 =nr
m1 =1
`=1
m1 =nr+1−` +1
m1 =nr−`
m1 =n1 +1
The terms corresponding to single term sums of the form m1 = n` , 1 ≤ ` ≤ r are merged
into the corresponding sums in ζN (k); then we recursively apply the same procedure to
products
ζm1 −1 (k`+1 , . . . , kr )ζm1 −1 (h2 , . . . , hs ),
which are of smaller length r − ` + s − 1 < r + s. Concerning the remaining sums we
simply interchange summations with the corresponding sums in ζN (k)ζN (h) and repeat
the same procedure to the arising products of finite multiple zeta functions, which are also
of smaller length. This proves the stated result.
Remark 2. Evidently, as remarked in [3], the relative order of the two strings is preserved,
but elements of the two strings may also be shoved together into a common slot (stuffing),
thereby reducing the depth.
[2]
[1]
Subsequently we use the notation `n = (`1 , `2 − `1 , . . . , n − `h−1 − 1), `n =
(`2 , , `3 − `2 , . . . , n − `h−1 − 1), and k = (k2 , . . . , kr ). For the simplification of S =
ζn∗1 ({1}n−1 )ζn1 −1 (k2 , . . . , kr ), as given in (3), we apply the stuffle identity of Lemma 1
[2]
[1]
to values ζn1 −1 (`n )ζn1 −1 (k) and ζn1 −1 (`n )ζn1 −1 (k). Hence, ξk1 ,...,kr (n) can be completely evaluated into finite sums of multiple zeta functions, which proves the second part
of Theorem 2. We state the explicit evaluation of ξk1 ,...,kr (n) in the following corollary.
Corollary 1. For k1 , . . . , kr , n ∈ N the function ξk1 ,...,kr (n) is given by
ξk1 ,...,kr (n) =
n−1
X
X
X
ζ(k1 + 1, f )
h=1 1≤`1 <`2 <···<`h−1 <n−1 f ∈stuffle(`[1] ,k)
n
+
n−1
X
X
X
ζ(k1 + 1 + `1 , f ),
h=1 1≤`1 <`2 <···<`h−1 <n−1 f ∈stuffle(`[2] ,k)
n
[1]
with respect to the notation `n
[2]
= (`1 , `2 −`1 , . . . , n−`h−1 −1), `n = (`2 , `3 −`2 , . . . , n−
`h−1 − 1), and k = (k2 , . . . , kr ).
R EFERENCES
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[2] T. Arakawa and M. Kaneko, Multiple zeta values, poly-Bernoulli numbers, and related zeta functions.
Nagoya Math. J. 153, 189–209, 1999.
[3] J. M. Borwein, D. M. Bradley, D. J. Broadhurst, and P. Lisonek, Special values of multidimensional polylogarithms, Trans. Amer. Math. Soc. 353, 907–941, 2001.
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Sums, Electronic Journal of Combinatorics 3, no. 1, #R23, 1996.
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[6] M. E. Hoffman, Multiple harmonic series. Pacific J. Math. vol. 152 no. 2, 275–290, 1992.
[7] S. Muneta, On some explicit evaluations of multiple star-zeta values, J. Number Theory 128, 2538–2548,
2008.
6
M. KUBA
[8] Y. Ohno, A Generalization of the Duality and Sum Formulas on the Multiple Zeta Values, Journal of Number
Theory 74, 39–43, 1999.
[9] Y. Ohno and J.-I. Okuda, On the sum formula for the q-analogue of non-strict multiple zeta values, Proceedings of the AMS, vol. 135, no. 10, 3029-3037, 2007.
[10] Y. Ohno and N. Wakabayashi, Cyclic sum of multiple zeta values, Acta Arith. 123, 289–295, 2006.
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M ARKUS K UBA , I NSTITUT F ÜR D ISKRETE M ATHEMATIK UND G EOMETRIE , T ECHNISCHE U NIVER W IEN , W IEDNER H AUPTSTR . 8-10/104, 1040 W IEN , AUSTRIA
E-mail address: [email protected]
SIT ÄT