Noncommutative algebra, multiple harmonic sums and applications in discrete probability (Q1025389)

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.