Vietnam J Math
DOI 10.1007/s10013-013-0047-x
Structure of Polyzetas and Lyndon Words
Hoang Ngoc Minh
Received: 11 October 2012 / Accepted: 17 August 2013
© Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore
2013
Abstract The effective construction of pairs of bases in duality for quasi-shuffle bialgebras
and the infinite factorizations by Lyndon words of noncommutative generating series allow
to prove that the algebra of polyzetas is graded by the weight and therefore lead to some
consequences for their arithmetical nature.
Keywords Algebraic computation · Combinatorial Hopf algebra · Drinfel’d associators ·
Free Lie algebra · Noncommutative symbolic computation · Polylogarithm · Polyzeta ·
Renormalization · Regularization · Special functions · Transcendence basis
Mathematics Subject Classification (2000) 05E · 11M32 · 37F25 · 68W30
1 Introduction
In 1734 Euler was the first to use his summation (also discovered afterwards, and inde-
pendently, by Mac-Laurin) to obtain the asymptotic expansion of the following classical
harmonic sums (for N,r ∈ N and N ≥ 1,r ≥ 2):
N
n=1
1
n
= log N + γ −
k−1
j =1
B
j
j
1
N
j
+ O
1
N
k
,
N
n=1
1
n
r
= ζ(r) −
k−1
j =r −1
B
j −r +1
j
j
r − 1
1
N
j
+ O
1
N
k
,
A Pierre Cartier, pour ses 80 ans.
Hoang Ngoc Minh (B )
Université Lille II, 1, Place Déliot, 59024 Lille, France
e-mail: hoang.ngocminh@lipn.univ-paris13.fr
Hoang Ngoc Minh
LIPN—UMR 7030, CNRS, 93430 Villetaneuse, France