Noncommutative algebra, multiple harmonic sums and applications in discrete probability

After having recalled some important results about combinatorics on words, like the existence of a basis for the shuffle algebras, we apply them to some special functions, the polylogarithms and to special numbers, the multiple harmonic sums . In the “good” cases, both objects converge (respectively, as z→1 and as N→+∞) to the same limit, the polyzêta . For the divergent cases, using the technologies of noncommutative generating series, we establish, by techniques “à la Hopf”, a theorem “à l’Abel”, involving the generating series of polyzêtas. This theorem enables one to give an explicit form to generalized Euler constants associated with the divergent harmonic sums, and therefore, to get a very efficient algorithm to compute the asymptotic expansion of any as N→+∞. Finally, we explore some applications of harmonic sums throughout the domain of discrete probabilities, for which our approach gives rise to exact computations, which can be then easily asymptotically evaluated.