Noncommutative algebra, multiple harmonic sums and applications in discrete probability
type: Item
instance of: scholarly article
title: Noncommutative algebra, multiple harmonic sums and applications in discrete probability
zbMATH Open document ID: 1179.11030
author: Christian Costermans
author: Vincel Hoang Ngoc Minh
published in: Journal of Symbolic Computation
publication date: 2009-06-18T00:00:00Z
review text: The paper surveys results about algebras on words equipped with shuffle products which are related to the multiple harmonic sums \[ H_{s_1,s_2,\dots,s_l}(N)=\sum_{N\geq n_1>n_2>\dots>n_l\geq1}\frac1{n_1^{s_1}n_2^{s_2}\dotsb n_l^{s_l}}, \] to the multiple polylogarithms \[ \text{Li}_{s_1,s_2,\dots,s_l}(z)=\sum_{n_1>n_2>\dots>n_l\geq1}\frac{z^{n_1}}{n_1^{s_1}n_2^{s_2}\dotsb n_l^{s_l}}, \] and to the multiple zeta values \[ \zeta(s_1,s_2,\dots,s_l)=\lim_{N\to\infty}H_{s_1,s_2,\dots,s_l}(N) =\lim_{z\to1}\text{Li}_{s_1,s_2,\dots,s_l}(z). \] The authors develop methods to compute the asymptotic behavior of \(H_{\mathbf s}(N)\) as \(N\to\infty\) (especially, in the divergent cases \(s_1=1\)) and give applications to evaluation of certain discrete probabilities.
reviewed by: Wadim Zudilin
Mathematics Subject Classification ID: 11M32
Mathematics Subject Classification ID: 05A05
Mathematics Subject Classification ID: 11G55
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 17A99
Mathematics Subject Classification ID: 33B30
Mathematics Subject Classification ID: 41A60
Mathematics Subject Classification ID: 60C05
zbMATH DE Number: 5566488
zbMATH Keywords: polylogarithm
zbMATH Keywords: multiple zeta value
zbMATH Keywords: harmonic sum
zbMATH Keywords: asymptotic analysis
zbMATH Keywords: discrete probability
MaRDI profile type: Publication
full work available at URL: j.jsc.2008.04.008
OpenAlex ID: W2015685619
cites work: On the variance of the number of maxima in random vectors and its applications
cites work: Maxima in hypercubes
cites work: On the Distribution of the Number of Admissible Points in a Vector Random Sample
cites work: Q5485340
cites work: Structure and asymptotic expansion of multiple harmonic sums
cites work: Q3575019
cites work: Q5749433
cites work: Q4855565
cites work: Q4524517
cites work: The hyperharmonic numbers and the phratry of the coupon collector
cites work: Q5187293
cites work: Q3839703
cites work: Q4453460
cites work: From the algebras of Riemann polyzetas to the algebra of Hurwitz polyzetas
cites work: Shuffle algebra and polylogarithms
cites work: The algebra of multiple harmonic series
cites work: Quasi-shuffle products
cites work: Algebraic Aspects of Multiple Zeta Values
cites work: Q3875087
cites work: Q4398780
cites work: Combinatorial variations on multidimensional quadtrees
cites work: On the distribution of the arity of the root of a \(d\)-dimensional quadtree
cites work: Q3575005
cites work: Q4328336
cites work: Duality between quasi-symmetric functions and the Solomon descent algebra
cites work: A natural ring basis for the shuffle algebra and an application to group schemes
cites work: Q3134850
DOI: 10.1016/J.JSC.2008.04.008
description: scientific article; zbMATH DE number 5566488