Multidimensional Polylogarithms

Algorithms Seminar

July 6, 1998

[summary by Hoang Ngoc Minh]

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## 1   Introduction

Recently, several extensions of polylogarithms, Euler sums (or multiple harmonic sums) and Riemann zeta functions have been introduced. These have arisen in number theory, knot theory, high-energy physics, analysis of quadtrees, control theory,...In this talk, the author presents the multidimensional polylogarithms and their special values [1, 2]. After definitions related to multidimensional polylogarithms (Section 2), results, conjectures and combinatorial aspects concerning unit Euler sums and unsigned Euler sums are discussed (Section 3). Integral representations are also pointed out to understand multidimensional polylogarithms (Section 4).

## 2   Definitions

Definition 1   The multidimensional polylogarithms (MDPs) are defined as follows
l
æ
è
 s1,...,sk b1,...,bk
ö
ø
=
 k Õ j=1
 å nj³1
b
 -nj j
(nj+...+nk)
 sj
.
k is the depth and s=s1+...+sk is the weight of l(
 s1,...,sk b1,...,bk
).
• When k=0, by convention l({})=1;
• When k=1, s is a positive integer and |b|³1, one get the usual polylogarithm
l
æ
è
 s b
ö
ø
=  å n³1
b  -n
ns
=Lis(1/b).
The classical Riemann zeta function is obtained in the special case where b=1.
• When k>1, let nj=åi=jkni and bj=Õi=1jai. Then
l
æ
è
 s1,...,sk b1,...,bk
ö
ø
=  å n1>...>nk>0
a  -n1 1
··· a  -nk k
n  s1 1
··· n  sk k
.
• If each aj=1 then these sums are called Euler sums;
• If each aj=±1 then they are called alternating Euler sums.
Definition 2   The unit Euler sum is defined as follows
µ(b1,...,bk)=l
æ
è
 1,...,1 b1,...,bk
ö
ø
=
 k Õ j=1
 å nj³1
b
 -nj j
(nj+...+nk)
.
Definition 3   The unsigned Euler sum is defined as follows
lb(s1,...,sk)=l
æ
è
 s1,...,sk b,...,b
ö
ø
=
 k Õ j=1
 å nj³1
b
 -nj
(nj+...+nk)
 sj
.
• When b=1, l1 is often called the unsigned Euler sum or multiple zeta value (MZV)
l1(s1,...,sk)=z(s1,...,sk) =  å n1>...>nk>0
1
n  s1 1
··· n  sk k
.
• When b=2, l2 represents an iterated sum extension of the polylogarithm with argument 1/2, and plays a crucial role in computing the MZVs.

## 3   Special Values of MDPs

Theorem 1   Let p and q satisfy 1/p+1/q=1. If in addition, p>1, or p£-1, then for any nonnegative integer k,
µ({p}k)=
(log q)k
k!
.
The proof is done by coefficient extraction in the generating function åk³0xkµ({p}k).
Theorem 2   Let Ar=Lir(1/2), Pr=(log 2)r/r!, Zr=(-1)rz(r). Then, for m³1,n³0
µ({-1}m,1,{-1}n)=(-1)m+1
 m å k=0
æ
è
 n+k k
ö
ø
Ak+n+1Pm-k +(-1)n+1
 n å k=0
æ
è
 m+k m
ö
ø
Zk+m+1Pn-k.
The proof of this theorem can be done via the duality principle (see Section 4).

For any nonnegative integer k, the following identities provide nested sum extensions of Euler's z(2),z(4),z(6) and z(8) evaluations, respectively
z({2}k)
=
2(2p)2k
(2k+1)!
æ
ç
ç
è
1
2
ö
÷
÷
ø
 2k+1
,
z({4}k)
=
4(2p)4k
(4k+2)!
æ
ç
ç
è
1
2
ö
÷
÷
ø
 2k+1
,
z({6}k)
=
6(2p)6k
(6k+3)!
,
z({8}k)
=
8(2p)8k
(8k+4)!
é
ê
ê
ê
ê
ê
ë
æ
ç
ç
è
1+
1
(2)1/2
ö
÷
÷
ø
 4k+2
+ æ
ç
ç
è
1-
1
(2)1/2
ö
÷
÷
ø
 4k+2
ù
ú
ú
ú
ú
ú
û
.
In general, for any positive integer n, e=eip/n, one has
 å k³0
(-1)kx2knz({2n}k)=
 n-1 Õ j=0
sin(p xej)
p xej
.
Theorem 3  [Zagier's conjecture ]
z({3,1}n)=4-nz({4}n)=
2p4n
(4n+2)!
.
Conjecture 1
z(2,{3,1}n)=4-n
 n å k=0
(-1)kz({4}n-k) é
ê
ê
ë
(4k+1)z(4n+2)-4
 k å j=1
z(4j-1)z(4k-4j+3) ù
ú
ú
û
.
In practice, one would like to know which unsigned Euler sums can be expressed in terms of lower depth sums. When the sum can be expressed, it is said to ``reduce''. Hoang Ngoc Minh and Michel Petitot have implemented in AXIOM an algorithm to reduce the MZVs via a table of Gröbner basis of these sums at fixed weight . Here, the authors also get the following
Theorem 4   For any positive integer k,
z(s1,...,sk)+(-1)kz(sk,...,s1)
reduces to lower depth MZVs.
The following theorem gives Crandall's recurrence for unsigned Euler sums z({s}k) and it can be proved by coefficient extraction in the generating function åk³0kxkz({s}k).
Theorem 5  [Crandall's recurrence]   For any nonnegative integer k and Â(s)>0,
kz({s}k)=
 k å j=1
(-1)j+1z(js)z({s}k-j).
For example
z({s}) =z(s),
z({s,s})
=
1
2
z2(s)-
1
2
z(2s),
z({s,s,s})
=
1
6
z3(s)-
1
2
z(s)z(2s)+
1
3
z(3s),...
Crandall's recurrence is also a special case of Newton's formula
kek=
 k å j=1
(-1)j+1pjek-j,   k³0,
relating the Elementary Symmetric Functions ek and and the Power-Sum Symmetric Functions pr,
ek=
 å j1>...>jr
x
 j1
··· x
 jr
,     pr=
 å r>0
xjr,
with indeterminates xj=1/js, er=z({s}r) and pr=z(rs).
Definition 4   Let s®=(s1,...,sk), t®=(t1,...,tr). The set stuffle(s®|t®) is defined as follows
1. (s1,...,sk,t1,...,tr)Îstuffle(s®|t®).
2. If (U,sn,tm,V) is in stuffle(s®|t®) then also are (U,tm,sn,V) and (U,sn+tm,V).
One also has
#stuffle (
 ® s
|
 ® t
)=
 r å j=0
æ
è
 k+j r
ö
ø
æ
è
 r j
ö
ø
=
 max(k,r) å j=0
æ
è
 k r
ö
ø
æ
è
 r j
ö
ø
2j.
Theorem 6  [Stuffle Identities ]
z(
 ® s
)z(
 ® t
)=

å
 ® u
Îstuffle (
 ® s
|
 ® t
)
z(
 ® u
).
For example
z(r,s)z(t)=z(r,s,t)+z(r,s+t)+z(r,t,s)+z(r+t,s)+z(t,r,s).

## 4   Integral Representations for MDPs

Let R1,...,Rk be disjoint sets of partitions of {1,...,k}. For each 1£ m£ n, let
rm=
 å iÎ Rm
si  and    dm=
 Õ iÎ Rm
bi.
From the gamma function identity
r-sG(s)= ó
õ
 ¥ 1
(log x)s-1x-r-1dx,   r,s>0.
one gets
Proposition 1
l
æ
è
 r1,...,rn d1,...,dn
ö
ø
= ì
ï
í
ï
î
 k Õ j=1
ó
õ
 ¥ 1
(log xj)
 sj-1
G(sj)
dxj
xj
ü
ï
ý
ï
þ
 n Õ m=1
æ
ç
ç
è
dm
 m Õ j=1
 Õ iÎ Rj
xi-1 ö
÷
÷
ø
 -1
.
For example, given a rational function on x and y, R(x,y). Let I(R) be the following partition integrals
I(R)= ó
õ
 ¥ 1
ó
õ
 ¥ 1
(log x)s-1(log y)t-1
G(s)G(t)
dxdy
xy R(x,y)
.
It follows that
l
æ
è
 s+t ab
ö
ø
=I(abxy-1),
l
æ
è
 s,t a,ab
ö
ø
=I[(ax-1)(abxy-1)],
l
æ
è
 t,s b,ab
ö
ø
=I[(by-1)(abxy-1)],
l
æ
è
 s a
ö
ø
l
æ
è
 t b
ö
ø
=I[(ax-1)(by-1)].
From the rational identity
1
(ax-1)(by-1)
=
1
abxy-1
æ
ç
ç
è
1
ax-1
+
1
by-1
+1 ö
÷
÷
ø
,
one gets
l
æ
è
 s a
ö
ø
l
æ
è
 t b
ö
ø
=l
æ
è
 s,t a,ab
ö
ø
+ l
æ
è
 t,s b,ab
ö
ø
+l
æ
è
 s+t ab
ö
ø
.
One can say that stuffle identities are equivalent to rational identities via partition integrals.
Definition 5   Given functions fj:[a,c]®R and the 1-forms Wj=fj(yj)dyj, the iterated integral over Wj are defined as follows
ó
õ
 c a
W1···Wn=
ì
ï
ï
í
ï
ï
î
1 if n=0,
ó
õ
 c a
f(y1) ó
õ
 y1 a
W2···Wn dy1
if n>0.
It turns out that MDPs have a convenient iterated integral representation in terms of 1-forms wb=dy/(y-b), i.e.
l
æ
è
 s1,...,sk b1,...,bk
ö
ø
=(-1)k ó
õ
 1 0
w
 s1-1 0
w
 b1
···w
 sk-1 0
w
 bk
.
By the iterated integral representation, Broadhurst has generalized the notion of duality principle for MZVs to include the relations between iterated integrals involving the sixth root of unity using the change of variable y|® 1-y at each level of integration . This principle generates an involution wb|®w1-b holding for any complex value b. For example
l
æ
è
 2,1 1,-1
ö
ø
= ó
õ
 1 0
w0w1w-1 = ó
õ
 1 0
w2w0w1=l
æ
è
 1,2 2,1
ö
ø
which is
 å n³1
1
n2
é
ê
ê
ë
 n-1 å k=1
(-1)k
k
ù
ú
ú
û
=
 å n³1
1
n2n
é
ê
ê
ë
 n-1 å k=1
2k
k2
ù
ú
ú
û
.
Several results can be similarly proved by using other transformations of variables in their integral representations. Here, the authors get
Theorem 7  [Cyclotomic]   Let n be a positive integer. Let b1,...,bk be arbitrary complex numbers, and let s1,...,sk be positive integers. Then
l
æ
è
 s1,...,sk b1n,...,bkn
ö
ø
=ns-k

å
e1,...,ekÎ{1,e
 2p i/n
,...,e
 2p(n-1)/n
}
l
æ
è
 s1,...,sk e1b1,...,ekbk
ö
ø
.
Theorem 8   Let s1,...,sk be nonnegative integers.
l
æ
è
 1+s1,...,1+sk -1,...,-1
ö
ø
=åµ æ
ç
ç
è
Catj=1k{-1}Cat
 sj i=1
{ei,j} ö
÷
÷
ø
 k Õ j=1
{-1}
 sj Õ i=1
ei,j,
where the sum is over all 2s sequences of signs (ei,j) with each ei,jÎ{1,-1} for all 1£ i£ sj,1£ j£ k, and Cat denotes string concatenation.

## References


Borwein (Jonathan M.), Bradley (David M.), and Broadhurst (David J.). -- Evaluations of k-fold Euler/Zagier sums: a compendium of results for arbitrary k. Electronic Journal of Combinatorics, vol. 4, n°2, 1997, pp. Research Paper 5, 21 pp. -- The Wilf Festschrift (Philadelphia, PA, 1996).


Borwein (Jonathan M.), Bradley (David M.), Broadhurst (David J.), and Petr (Lisonek). -- Special values of multidimensional polylogarithms. -- Research report n°98-106, CECM, 1998. Available at the URL http://www.cecm.sfu.ca/preprints/1998pp.html.


Broadhurst (D. J.). -- Massive 3-loop Feynman Diagrams Reducible to SC* primitives of algebras of the sixth root of unity. -- Technical Report n°OUT-4102-72, hep-th/9803091, Open University, 1998.


Hoffman (Michael E.). -- The algebra of multiple harmonic series. Journal of Algebra, vol. 194, n°2, 1997, pp. 477--495.


Minh (Hoang Ngoc) and Petitot (M.). -- Lyndon words, polylogarithmic functions and the Riemann z function. -- Preprint.


Zagier (Don). -- Values of zeta functions and their applications. In et al. (A. Joseph) (editor), Proceedings of the First European Congress of Mathematics, Paris. vol. II, pp. 497--512. -- Birkhäuser Verlag, 1994. (Progress in Mathematics, volume 120.).

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