Multidimensional Polylogarithms (original) (raw)

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 (Section2), 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

_k is the depth and s=s 1 +...+s k is the weight_of l (

) .

• When _k_=0, by convention l({})=1;
• When _k_=1, s is a positive integer and |b|�1, one get the usual _polylogarithm_The classical Riemann zeta function is obtained in the special case where _b_=1.
• When _k_>1, let n _j_=�_i_=j k_n_i and b j_=�_i_=1_j a i. Then
  • 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

Definition 3 The unsigned Euler sum is defined as follows

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

  l1(_s_1,...,s k)=z(_s_1,...,s 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 ,

The proof is done by coefficient extraction in the generating function �k_�0_x _k_�({p}k).

Theorem 2 Let _A r = Li r (1/2), P r = ( log 2) r / r! , Z r =(-1) r z(r). Then, for _m�1,n�0

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

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

Theorem 3 [Zagier's conjecture [6]]

Conjecture 1

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 Gr�bner basis of these sums at fixed weight [5]. Here, the authors also get the following

Theorem 4 For any positive integer _k ,

z(s 1 ,...,s k )+(-1) k z(s k ,...,s 1 )

reduces to lower depth MZVs.

The following theorem gives Crandall's recurrence for unsigned Euler sumsz({s}k) and it can be proved by coefficient extraction in the generating function�k_�0_kx _k_z({s}k).

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

For example

Crandall's recurrence is also a special case of Newton's formula relating the Elementary Symmetric Functions e k and and the Power-Sum Symmetric Functions p r, with indeterminates x _j_=1/j s,e _r_=z({s}r) and p _r_=z(rs).

Definition 4 Let _s � =(s 1 ,...,s k ) , _t � =(t 1 ,...,t r ) . The set stuffle (s � |t � ) is defined as follows

1. (s 1 ,...,s k ,t 1 ,...,t r )� stuffle (s � |t � ) .
2. 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

Theorem 6 [Stuffle Identities [4]]

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 _R_1,...,R k be disjoint sets of partitions of {1,...,k}. For each 1� _m_� n, let From the gamma function identity

one gets

Proposition 1

| l | = | ����� | �� | | ����� | ���� | _d m | _x i -1 | ���� | . | | --- | ---- | ------- | ---- | | ------- | ------ | -------- | -------------- | ------ | --- |

For example, given a rational function on x and y, R(x,y). Let I(R) be the following partition integrals

| I(R)= | �� | �� | | (log x)_s_-1(log y)_t_-1 G(s)G(t) | . | | --------- | -- | -- | | ------------------------------------------- | - |

It follows that

From the rational identity one gets 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

It turns out that MDPs have a convenient iterated integral representation in terms of 1-forms w_b_=dy/(y_-b), i.e. 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 involutionw_b|�w1-b holding for any complex value b. For example

which is 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 _b 1 ,...,b k be arbitrary complex numbers, and let s 1 ,...,s k be positive integers. Then

Theorem 8 Let _s 1 ,...,s k be nonnegative integers.

where the sum is over all 2 s sequences of signs ( e i,j ) with each e i,j �{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. -- Birkh�user Verlag, 1994. (Progress in Mathematics, volume 120.).

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