Noncommutative algebra, multiple harmonic sums and applications in discrete probability
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.
scientific article; zbMATH DE number 5566488
scientific article