## Special Values of Multiple Polylogarithms

Venue: | Sém. Bourbaki, 53 e année, 2000–2001, n ◦ 885, Mars 2001; Astéisque 282 (2002 |

Citations: | 62 - 18 self |

### BibTeX

@INPROCEEDINGS{Borwein_specialvalues,

author = {Jonathan M. Borwein and David M. Bradley and David J. Broadhurst and Petr and Lison Ěk},

title = {Special Values of Multiple Polylogarithms},

booktitle = {Sém. Bourbaki, 53 e année, 2000–2001, n ◦ 885, Mars 2001; Astéisque 282 (2002},

year = {},

pages = {907--941}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. Historically, the polylogarithm has attracted specialists and nonspecialists alike with its lovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within the past decade, have arisen in combinatorics, knot theory and high-energy physics. More recently, we have been forced to consider multidimensional extensions encompassing the classical polylogarithm, Euler sums, and the Riemann zeta function. Here, we provide a general framework within which previously isolated results can now be properly understood. Applying the theory developed herein, we prove several previously conjectured evaluations, including an intriguing conjecture of Don Zagier. 1.

### Citations

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Citation Context ...e of the q-binomial theorem, usually expressed in the more familiar form [32] as (\Gammazq; q) 1 = 1 X k=0 q k(k+1)=2 (q; q) k z k : The case k = 1, b 1 = 2, s 1 = \Gamman of (2.6) yields the numbers =-=[63]-=- (A000629) ffi(\Gamman) =s2 (\Gamman) = 1 X k=1 k n 2 k = Li \Gamman ( 1 2 ); 0sn 2 Z; (2.8) which enumerate [45] the combinations of a simplex lock having n buttons, and which satisfy the recurrence ... |

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Citation Context ...vation of Markett's formula [51] for i(m; 1; 1), ms2. Thus, the complete reducibility of i(m+2; f1g n ) is a simple consequence of the instance (6.5) of Gauss's 2 F 1 hypergeometric summation theorem =-=[1, 3, 62]-=-. Wenchang Chu [19] has elaborated on this idea, applying additional hypergeometric summation theorems to evaluate a wide variety of depth-2 sums, including nonlinear (cf. [31]) sums. It would be inte... |

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Citation Context ...nce ffi(\Gamman) = 1 + n\Gamma1 X j=0 ` n j ' ffi(\Gammaj ); 1sn 2 Z: Also, from the exponential generating function 1 X n=0 ffi(\Gamman) x n n! = e x 2 \Gamma e x = 2 2 \Gamma e x \Gamma 1; we infer =-=[36, 64]-=- that for ns1, 1 2 ffi(\Gamman) also counts ffl the number of ways of writing a sum on n indices; ffl the number of functions f : f1; 2; : : : ; ng ! f1; 2; : : : ; ng such that if j is in the range o... |

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Citation Context ...ion (6.6) and the result (6.10) in the form ffi(r) = Li r ( 1 2 ) and ffi(f1g r ) = (log 2) r =r!, respectively. SPECIAL VALUES OF MULTIPLE POLYLOGARITHMS 23 Example 7.1. Putting n = 1 in (7.3) gives =-=[5]-=- i(3) = 1 X n=1 1 n 3 = 1 12 2 log(2) + 1 X n=1 1 2 n n 2 n X j=1 1 j : (7.4) In fact, formula (7.3) is non-trivial even when n = 0. Putting n = 0 in (7.3) gives the classical evaluation of the diloga... |

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99 |
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Citation Context ...ral representation (4.2) yields a weight-dimensional iterated integral. The advantage here is that the rational function comprising the integrand is particularly simple. We use the notation of Kassel =-=[44]-=- for iterated integrals. For j = 1; 2; : : : ; n, let f j : [a; c] ! R and\Omega j := f j (y j ) dy j . Then Z c a \Omega 1\Omega 2 \Delta \Delta \Delta\Omega n := n Y j=1 Z y j\Gamma1 a f j (y j ) dy... |

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Citation Context ... cases of certain formulae for alternating sums must be treated separately unless running product notation is used. Theorem 8.5 with n = 0 (Section 8) provides an example of this. Don Zagier (see eg. =-=[69]) has-=- argued persuasively in favour of studying special values of zeta functions at integer arguments, as these values "often seem to dictate the most important properties of the objects to which the ... |

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Citation Context ...X n1?\Delta\Delta\Delta?n k ?0 k Y j=1 n \Gammas j j x j n j : (1.4) With each x j = 1, these latter sums (sometimes called "Euler sums"), have been studied previously at various levels of g=-=enerality [2, 6, 7, 9, 13, 14, 15, 16, 31, 38, 39, 42, 51, 59]-=-, the case k = 2 going back to Euler [27]. Recently, Euler sums have arisen in combinatorics (analysis of quad-trees [30, 46] and of lattice reduction algorithms [23]), knot theory [14, 15, 16, 47], a... |

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Citation Context ...; c 4 ) is greater than 10 50 . This result can be proved computationally in a mere 0.2 seconds on a DEC Alpha workstation using D. Bailey's fast implementation of the integer relation algorithm PSLQ =-=[29]-=-, once we know the four input values at the precision of 200 decimal digits. Such evaluation poses no obstacle to our fast method of evaluating polylogs using the Holder convolution (see Section 7). 3... |

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Citation Context ...y particle physics [13] (quantum field theory). There is also quite sophisticated work relating polylogarithms and their generalizations to arithmetic and algebraic geometry, and to algebraic Ktheory =-=[4, 17, 18, 33, 34, 35, 66, 67, 68]-=-. In view of these recent applications and the well-known fact that the classical polylogarithm (1.2) often arises in physical problems via the multiple integration of rational forms, one might expect... |

46 |
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Citation Context ... Example 6.2. MZV duality (6.4) gives Euler's evaluation i(2; 1) = i(3); as well as the generalizations i(f2; 1g n ) = i(f3g n ); and i(2; f1g n ) = i(n + 2), valid for all nonnegative integers n. In =-=[60]-=- a beautiful extension of MZV duality (6.4) is given, which also subsumes the so-called sum formula X n j ?ffi j;1 N =\Sigma j n j i(n 1 ; n 2 ; : : : ; n k ) = i(N); conjectured independently by C. M... |

44 |
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Citation Context ...hese latter sums (sometimes called "Euler sums"), have been studied previously at various levels of generality [2, 6, 7, 9, 13, 14, 15, 16, 31, 38, 39, 42, 51, 59], the case k = 2 going back=-= to Euler [27]-=-. Recently, Euler sums have arisen in combinatorics (analysis of quad-trees [30, 46] and of lattice reduction algorithms [23]), knot theory [14, 15, 16, 47], and high-energy particle physics [13] (qua... |

42 | Massive 3-loop Feynman diagrams reducible to SC ∗ primitives of algebras of the sixth root of unity, Eur
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Citation Context ...The duality principle states that two MZVs coincide whenever their argument strings are dual to each other, and (as noted by Zagier [69]) follows readily from the iterated integral representation. In =-=[12]-=-, Broadhurst generalized the notion of duality to include relations between iterated integrals involving the sixth root of unity; here we allow arbitrary complex values of b j . Thus, we find that the... |

38 |
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Citation Context ...vation of Markett's formula [51] for i(m; 1; 1), ms2. Thus, the complete reducibility of i(m+2; f1g n ) is a simple consequence of the instance (6.5) of Gauss's 2 F 1 hypergeometric summation theorem =-=[1, 3, 62]-=-. Wenchang Chu [19] has elaborated on this idea, applying additional hypergeometric summation theorems to evaluate a wide variety of depth-2 sums, including nonlinear (cf. [31]) sums. It would be inte... |

30 |
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Citation Context ...y particle physics [13] (quantum field theory). There is also quite sophisticated work relating polylogarithms and their generalizations to arithmetic and algebraic geometry, and to algebraic Ktheory =-=[4, 17, 18, 33, 34, 35, 66, 67, 68]-=-. In view of these recent applications and the well-known fact that the classical polylogarithm (1.2) often arises in physical problems via the multiple integration of rational forms, one might expect... |

25 |
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Citation Context ...X n1?\Delta\Delta\Delta?n k ?0 k Y j=1 n \Gammas j j x j n j : (1.4) With each x j = 1, these latter sums (sometimes called "Euler sums"), have been studied previously at various levels of g=-=enerality [2, 6, 7, 9, 13, 14, 15, 16, 31, 38, 39, 42, 51, 59]-=-, the case k = 2 going back to Euler [27]. Recently, Euler sums have arisen in combinatorics (analysis of quad-trees [30, 46] and of lattice reduction algorithms [23]), knot theory [14, 15, 16, 47], a... |

25 |
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Citation Context ...elta \Delta Z 1 1 k Y j=1 (log x j ) s j \Gamma1 dx j \Gamma(s j ) \Gamma b j Q j i=1 x i \Gamma 1 \Delta x j ; (4.2) SPECIAL VALUES OF MULTIPLE POLYLOGARITHMS 9 which generalizes Crandall's integral =-=[20]-=- for i(s 1 ; : : : ; s k ). An equivalent formulation of (4.2) is ` s 1 ; : : : ; s k b 1 ; : : : ; b k ' = Z 1 0 \Delta \Delta \Delta Z 1 0 k Y j=1 u s j \Gamma1 j du j \Gamma(s j )(b j exp \Gamma P ... |

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Citation Context ...1sj Xsj =0 A( j ) oe ffi(s \Gammasr \Gamma 1); s 2 C; wheresj := n j +sj \Gamma1 + 1; A( j ) := 1sj + 1 `sjsj ' Bsj \Gamma j ;s0 := \Gamma1: 7. The H older Convolution Richard Crandall [21] (see also =-=[22]-=-) describes a practical method for fast evaluation of MZVs. Here, we develop an entirely different approach which is based on the fact that any multiple polylogarithm can be expressed as a convolution... |

17 |
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Citation Context ...te that ` s b ' = 1 X =1 1 s b = Li s ` 1 b ' (1.2) is the usual polylogarithm [49, 50] when s is a positive integer and jbjs1. Of course, the polylogarithm (1.2) reduces to the Riemann zeta function =-=[26, 43, 65]-=- i(s) = 1 X =1 1 s ; !(s) ? 1; (1.3) when b = 1. More generally, for any k ? 0 the substitution n j = P k i=jsi shows that our multiple polylogarithm (1.1) is related to Goncharov's [35] by the equati... |

13 |
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Citation Context ...is given, which also subsumes the so-called sum formula � ζ(n1,n2,... ,nk) =ζ(N), nj>δj,1 N=Σjnj conjectured independently by C. Moen [38] and M. Schmidt [51], and subsequently proved by A. Granville =-=[37]-=-. We refer the reader to Dr. Ohno’s article for details. The duality principle has an enticing converse, namely that two MZVs with distinct argument strings are equal only if the argument strings are ... |

12 |
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Citation Context ...mes the so-called sum formula X n j ?ffi j;1 N =\Sigma j n j i(n 1 ; n 2 ; : : : ; n k ) = i(N); conjectured independently by C. Moen [38] and M. Schmidt [51], and subsequently proved by A. Granville =-=[37]-=-. We refer the reader to Dr. Ohno's article for details. The duality principle has an enticing converse, namely that two MZVs with distinct argument strings are equal only if the argument strings are ... |

12 |
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Citation Context ...l refer to as multiple polylogarithms. When k = 0, we define (fg) := 1, where fg denotes the empty string. When k = 1, note that ` s b ' = 1 X =1 1 s b = Li s ` 1 b ' (1.2) is the usual polylogarithm =-=[49, 50]-=- when s is a positive integer and jbjs1. Of course, the polylogarithm (1.2) reduces to the Riemann zeta function [26, 43, 65] i(s) = 1 X =1 1 s ; !(s) ? 1; (1.3) when b = 1. More generally, for any k ... |

11 |
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11 |
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Citation Context ...2 = 1 X n=0 2 4n t 4n (4n + 2)! : Remark 11.2. The proof is Zagier's modification of Broadhurst's, based on the extensive empirical work begun in [7]. 11.3. Generalizations of Zagier's Conjecture. In =-=[8]-=- we give an alternative (combinatorial) proof of Zagier's conjecture, based on combinatorial manipulations of the iterated integral representations of MZVs (see Sections 4.2 and 5.4). Using the same t... |

9 |
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Citation Context ...la [51] for i(m; 1; 1), ms2. Thus, the complete reducibility of i(m+2; f1g n ) is a simple consequence of the instance (6.5) of Gauss's 2 F 1 hypergeometric summation theorem [1, 3, 62]. Wenchang Chu =-=[19]-=- has elaborated on this idea, applying additional hypergeometric summation theorems to evaluate a wide variety of depth-2 sums, including nonlinear (cf. [31]) sums. It would be interesting to know if ... |

8 |
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Citation Context ...t various levels of generality [2, 6, 7, 9, 13, 14, 15, 16, 31, 38, 39, 42, 51, 59], the case k = 2 going back to Euler [27]. Recently, Euler sums have arisen in combinatorics (analysis of quad-trees =-=[30, 46]-=- and of lattice reduction algorithms [23]), knot theory [14, 15, 16, 47], and high-energy particle physics [13] (quantum field theory). There is also quite sophisticated work relating polylogarithms a... |

7 |
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Citation Context ...1)=2 (q; q) k z k : The case k = 1, b 1 = 2, s 1 = \Gamman of (2.6) yields the numbers [63] (A000629) ffi(\Gamman) =s2 (\Gamman) = 1 X k=1 k n 2 k = Li \Gamman ( 1 2 ); 0sn 2 Z; (2.8) which enumerate =-=[45]-=- the combinations of a simplex lock having n buttons, and which satisfy the recurrence ffi(\Gamman) = 1 + n\Gamma1 X j=0 ` n j ' ffi(\Gammaj ); 1sn 2 Z: Also, from the exponential generating function ... |

7 |
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Citation Context ...38, 39, 42, 51, 59], the case k = 2 going back to Euler [27]. Recently, Euler sums have arisen in combinatorics (analysis of quad-trees [30, 46] and of lattice reduction algorithms [23]), knot theory =-=[14, 15, 16, 47]-=-, and high-energy particle physics [13] (quantum field theory). There is also quite sophisticated work relating polylogarithms and their generalizations to arithmetic and algebraic geometry, and to al... |

7 |
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Citation Context ...y particle physics [13] (quantum field theory). There is also quite sophisticated work relating polylogarithms and their generalizations to arithmetic and algebraic geometry, and to algebraic Ktheory =-=[4, 17, 18, 33, 34, 35, 66, 67, 68]-=-. In view of these recent applications and the well-known fact that the classical polylogarithm (1.2) often arises in physical problems via the multiple integration of rational forms, one might expect... |

5 |
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Citation Context ...l the number of asymmetric generalized weak orders on f1; 2; : : : ; ng; ffl the number of ordered partitions (preferential arrangements) of f1; 2; : : : ; ng. The numbers 1 2 ffi(\Gamman) also arise =-=[24]-=- in connection with certain constants related to the Laurent coefficients of the Riemann zeta function. See [63] (A000670) for additional references. 3. Reductions Given the multiple polylogarithm (1.... |

5 |
Factoring Polynomials with
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Citation Context ...ted. The exact status of the EZ Face is at any moment documented at its "Definitions" and "Using EZ-Face" pages. In addition to the functions z and zp, the lindep function, based o=-=n the LLL algorithm [48]-=- for discovering integer relations [10] satisfied by a vector of real numbers, can be called. An integer relation for a vector of real numbers (x 1 ; : : : ; xn ) is a non-zero integer vector (c 1 ; :... |

4 |
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Citation Context ...3, 14, 15, 16, 31, 38, 39, 42, 51, 59], the case k = 2 going back to Euler [27]. Recently, Euler sums have arisen in combinatorics (analysis of quad-trees [30, 46] and of lattice reduction algorithms =-=[23]-=-), knot theory [14, 15, 16, 47], and high-energy particle physics [13] (quantum field theory). There is also quite sophisticated work relating polylogarithms and their generalizations to arithmetic an... |

4 |
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Citation Context ...te that ` s b ' = 1 X =1 1 s b = Li s ` 1 b ' (1.2) is the usual polylogarithm [49, 50] when s is a positive integer and jbjs1. Of course, the polylogarithm (1.2) reduces to the Riemann zeta function =-=[26, 43, 65]-=- i(s) = 1 X =1 1 s ; !(s) ? 1; (1.3) when b = 1. More generally, for any k ? 0 the substitution n j = P k i=jsi shows that our multiple polylogarithm (1.1) is related to Goncharov's [35] by the equati... |

4 |
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4 |
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Citation Context ...te that ` s b ' = 1 X =1 1 s b = Li s ` 1 b ' (1.2) is the usual polylogarithm [49, 50] when s is a positive integer and jbjs1. Of course, the polylogarithm (1.2) reduces to the Riemann zeta function =-=[26, 43, 65]-=- i(s) = 1 X =1 1 s ; !(s) ? 1; (1.3) when b = 1. More generally, for any k ? 0 the substitution n j = P k i=jsi shows that our multiple polylogarithm (1.1) is related to Goncharov's [35] by the equati... |

3 |
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of Multiple Zeta Values with Positive Knots via Feynman Diagrams up to 9
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Citation Context ...1 ) = ae r Y j=1sj Xsj =0 A( j ) oe ffi(s \Gammasr \Gamma 1); s 2 C; wheresj := n j +sj \Gamma1 + 1; A( j ) := 1sj + 1 `sjsj ' Bsj \Gamma j ;s0 := \Gamma1: 7. The H older Convolution Richard Crandall =-=[21]-=- (see also [22]) describes a practical method for fast evaluation of MZVs. Here, we develop an entirely different approach which is based on the fact that any multiple polylogarithm can be expressed a... |

3 |
Basic Hypergeometric Series, with a forward by Richard Askey
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Citation Context ...experts in the field of basic hypergeometric series will recognize as a qanalogue of the exponential function and a special case of the q-binomial theorem, usually expressed in the more familiar form =-=[32]-=- as (\Gammazq; q) 1 = 1 X k=0 q k(k+1)=2 (q; q) k z k : The case k = 1, b 1 = 2, s 1 = \Gamman of (2.6) yields the numbers [63] (A000629) ffi(\Gamman) =s2 (\Gamman) = 1 X k=1 k n 2 k = Li \Gamman ( 1 ... |

3 |
Combinatorial Enumeration, with a forward by Gian Carlo-Rota
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Citation Context ...nce ffi(\Gamman) = 1 + n\Gamma1 X j=0 ` n j ' ffi(\Gammaj ); 1sn 2 Z: Also, from the exponential generating function 1 X n=0 ffi(\Gamman) x n n! = e x 2 \Gamma e x = 2 2 \Gamma e x \Gamma 1; we infer =-=[36, 64]-=- that for ns1, 1 2 ffi(\Gamman) also counts ffl the number of ways of writing a sum on n indices; ffl the number of functions f : f1; 2; : : : ; ng ! f1; 2; : : : ; ng such that if j is in the range o... |

3 |
de Dirichlet d’ordre n et de Paramètre t
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Citation Context ...2 of Section 11.2) uses little more than the combinatorial properties of shuffles [8]. In addition, both shuffles and stuffles have featured in the investigations of other authors in related contexts =-=[39, 40, 41, 52, 53, 54, 55, 56, 57, 58, 61]-=-. 6. Duality In [38], Hoffman defines an involution on strings s 1 ; : : : ; s k . The involution coincides with a notion we refer to as duality. The duality principle states that two MZVs coincide wh... |

3 |
Ngoc Minh and Michel Petitot, Mots de Lyndon: Générateurs de Relations entre les Polylogarithmes de Nielsen, presented at
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Citation Context ...2 of Section 11.2) uses little more than the combinatorial properties of shuffles [8]. In addition, both shuffles and stuffles have featured in the investigations of other authors in related contexts =-=[39, 40, 41, 52, 53, 54, 55, 56, 57, 58, 61]-=-. 6. Duality In [38], Hoffman defines an involution on strings s 1 ; : : : ; s k . The involution coincides with a notion we refer to as duality. The duality principle states that two MZVs coincide wh... |

3 |
words, polylogarithms and the Riemann function. submited to Discret Maths
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Citation Context ...pth MZVs, and which was subsequently proved. Observe that the combined weight of each term in the reduction (3.1) is preserved. The easiest proof of (3.1) uses Minh and Petitot's basis of order eight =-=[55]-=-. Broadhurst also noted that although i(4; 2; 4; 2) is apparently irreducible in terms of lower depth MZVs, we have the conjectured 1 weight-12 reduction i(4; 2; 4; 2) ? = \Gamma 1024 27 (9\Gamma; 3) ... |

3 |
des Polylogarithmes par les Séries Génératrices, presented at FPSAC
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Citation Context ...2 of Section 11.2) uses little more than the combinatorial properties of shuffles [8]. In addition, both shuffles and stuffles have featured in the investigations of other authors in related contexts =-=[39, 40, 41, 52, 53, 54, 55, 56, 57, 58, 61]-=-. 6. Duality In [38], Hoffman defines an involution on strings s 1 ; : : : ; s k . The involution coincides with a notion we refer to as duality. The duality principle states that two MZVs coincide wh... |