Multidimensional Polylogarithms

Multidimensional Polylogarithms

David M. Bradley

Dalhousie University, Canada

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

æ
è

s₁,...,s_k

b₁,...,b_k

ö
ø

j=1

n_j³1

-n_j

(n_j+...+n_k)

s_j

k is the depth and s=s₁+...+s_k is the weight of l(

s₁,...,s_k

b₁,...,b_k

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

n^s
=Li_s(1/b).
The classical Riemann zeta function is obtained in the special case where b=1.
When k>1, let n_j=å_i=j^kn_i and b_j=Õ_i=1^ja_i. Then

l

æ
è

s₁,...,s_k

b₁,...,b_k
ö
ø
=

å

n₁>...>n_k>0

a

-n₁

1
··· a

-n_k

k

n

s₁

1
··· n

s_k

k

.
- If each a_j=1 then these sums are called Euler sums;
- If each a_j=±1 then they are called alternating Euler sums.

Definition 2 The unit Euler sum is defined as follows

µ(b₁,...,b_k)=l

æ
è

1,...,1

b₁,...,b_k

ö
ø

j=1

n_j³1

-n_j

(n_j+...+n_k)

Definition 3 The unsigned Euler sum is defined as follows

l_b(s₁,...,s_k)=l

æ
è

s₁,...,s_k

b,...,b

ö
ø

j=1

n_j³1

-n_j

(n_j+...+n_k)

s_j

When b=1, l₁ is often called the unsigned Euler sum or multiple zeta value (MZV)

l₁(s₁,...,s_k)=z(s₁,...,s_k) =

å

n₁>...>n_k>0

1

n

s₁

1
··· n

s_k

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

The proof is done by coefficient extraction in the generating function å_k³0x^kµ({p}^k).

Theorem 2 Let A_r=Li_r(1/2), P_r=(log 2)^r/r!, Z_r=(-1)^rz(r). Then, for m³1,n³0

µ({-1}^m,1,{-1}ⁿ)=(-1)^m+1

k=0

æ
è

n+k

ö
ø

A_k+n+1P_m-k +(-1)ⁿ⁺¹

k=0

æ
è

m+k

ö
ø

Z_k+m+1P_n-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

ö
÷
÷
ø

2k+1

z({4}^k)

4(2p)^4k

(4k+2)!

æ
ç
ç
è

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

(2)^1/2

ö
÷
÷
ø

4k+2

æ
ç
ç
è

(2)^1/2

ö
÷
÷
ø

4k+2

ù
ú
ú
ú
ú
ú
û

In general, for any positive integer n, e=e^ip/n, one has

k³0

(-1)^kx^2knz({2n}^k)=

n-1

j=0

sin(p xe^j)

p xe^j

Theorem 3 [Zagier's conjecture [6]]

z({3,1}ⁿ)=4^-nz({4}ⁿ)=

2p⁴ⁿ

(4n+2)!

Conjecture 1

z(2,{3,1}ⁿ)=4^-n

k=0

(-1)^kz({4}^n-k)

é
ê
ê
ë

(4k+1)z(4n+2)-4

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 [5]. Here, the authors also get the following

Theorem 4 For any positive integer k,

z(s₁,...,s_k)+(-1)^kz(s_k,...,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 å_k³0kx^kz({s}^k).

Theorem 5 [Crandall's recurrence] For any nonnegative integer k and Â(s)>0,

kz({s}^k)=

j=1

(-1)^j+1z(js)z({s}^k-j).

For example

z({s})

=z(s),

z({s,s})

z²(s)-

z(2s),

z({s,s,s})

z³(s)-

z(s)z(2s)+

z(3s),...

Crandall's recurrence is also a special case of Newton's formula

ke_k=

j=1

(-1)^j+1p_je_k-j, k³0,

relating the Elementary Symmetric Functions e_k and and the Power-Sum Symmetric Functions p_r,

e_k=

j₁>...>j_r

j₁

··· x

j_r

, p_r=

r>0

x_j^r,

with indeterminates x_j=1/j^s, e_r=z({s}^r) and p_r=z(rs).

Definition 4 Let s^®=(s₁,...,s_k), t^®=(t₁,...,t_r). The set stuffle(s^®|t^®) is defined as follows

(s₁,...,s_k,t₁,...,t_r)Îstuffle(s^®|t^®).
If (U,s_n,t_m,V) is in stuffle(s^®|t^®) then also are (U,t_m,s_n,V) and (U,s_n+t_m,V).

One also has

#stuffle

(

j=0

æ
è

k+j

ö
ø

æ
è

ö
ø

max(k,r)

j=0

æ
è

ö
ø

æ
è

ö
ø

2^j.

Theorem 6 [Stuffle Identities [4]]

)z(

Îstuffle

(

)

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,...,R_k be disjoint sets of partitions of {1,...,k}. For each 1£ m£ n, let

r_m=

iÎ R_m

s_i and

d_m=

iÎ R_m

b_i.

From the gamma function identity

r^-sG(s)=

ó
õ

(log x)^s-1x^-r-1dx, r,s>0.

one gets

Proposition 1

æ
è

r₁,...,r_n

d₁,...,d_n

ö
ø

ì
ï
í
ï
î

j=1

ó
õ

(log

x_j)

s_j-1

G(s_j)

dx_j

x_j

ü
ï
ý
ï
þ

m=1

æ
ç
ç
è

d_m

j=1

iÎ R_j

x_i-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

(log x)^s-1(log y)^t-1

G(s)G(t)

dxdy

xy R(x,y)

It follows that

æ
è

s+t

ö
ø

=I(abxy-1),

æ
è

s,t

a,ab

ö
ø

=I[(ax-1)(abxy-1)],

æ
è

t,s

b,ab

ö
ø

=I[(by-1)(abxy-1)],

æ
è

ö
ø

æ
è

ö
ø

=I[(ax-1)(by-1)].

From the rational identity

(ax-1)(by-1)

abxy-1

æ
ç
ç
è

ax-1

by-1

ö
÷
÷
ø

one gets

æ
è

ö
ø

æ
è

ö
ø

æ
è

s,t

a,ab

ö
ø

+ l

æ
è

t,s

b,ab

ö
ø

æ
è

s+t

ö
ø

One can say that stuffle identities are equivalent to rational identities via partition integrals.

Definition 5 Given functions f_j:[a,c]®R and the 1-forms W_j=f_j(y_j)dy_j, the iterated integral over W_j are defined as follows

ó
õ

W₁···W_n=

ì
ï
ï
í
ï
ï
î

if n=0,

ó
õ

f(y₁)

ó
õ

y₁

W₂···W_n dy₁

if n>0.

It turns out that MDPs have a convenient iterated integral representation in terms of 1-forms w_b=dy/(y-b), i.e.

æ
è

s₁,...,s_k

b₁,...,b_k

ö
ø

=(-1)^k

ó
õ

s₁-1

b₁

···w

s_k-1

b_k

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 [3]. This principle generates an involution w_b|®w_1-b holding for any complex value b. For example

æ
è

2,1

1,-1

ö
ø

ó
õ

w₀w₁w_-1 =

ó
õ

w₂w₀w₁=l

æ
è

1,2

2,1

ö
ø

which is

n³1

n²

é
ê
ê
ë

n-1

k=1

(-1)^k

ù
ú
ú
û

n³1

n2ⁿ

é
ê
ê
ë

n-1

k=1

2^k

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,...,b_k be arbitrary complex numbers, and let s₁,...,s_k be positive integers. Then

æ
è

s₁,...,s_k

b₁ⁿ,...,b_kⁿ

ö
ø

=n^s-k

e₁,...,e_kÎ{1,e

2p i/n

,...,e

2p(n-1)/n

}

æ
è

s₁,...,s_k

e₁b₁,...,e_kb_k

ö
ø

Theorem 8 Let s₁,...,s_k be nonnegative integers.

æ
è

1+s₁,...,1+s_k

-1,...,-1

ö
ø

=åµ

æ
ç
ç
è

Cat_j=1^k{-1}Cat

s_j

i=1

{e_i,j}

ö
÷
÷
ø

j=1

{-1}

s_j

i=1

e_i,j,

where the sum is over all 2^s sequences of signs (e_i,j) with each e_i,jÎ{1,-1} for all 1£ i£ s_j,1£ j£ k, and Cat denotes string concatenation.

References

[1]: 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).
[2]: 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.
[3]: 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.
[4]: Hoffman (Michael E.). -- The algebra of multiple harmonic series. Journal of Algebra, vol. 194, n°2, 1997, pp. 477--495.
[5]: Minh (Hoang Ngoc) and Petitot (M.). -- Lyndon words, polylogarithmic functions and the Riemann z function. -- Preprint.
[6]: 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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