Combinatorial variations on multidimensional quadtrees
Journal of Combinatorial Theory Series A
On the distribution of the root arity of a d dimensional hyperquaternary branching
Proceedings of the 4th conference on Formal power series and algebraic combinatorics
An introduction to the analysis of algorithms
An introduction to the analysis of algorithms
Journal of Algebraic Combinatorics: An International Journal
Structure and asymptotic expansion of multiple harmonic sums
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
Random Structures & Algorithms
Schützenberger's factorization on q?stuffle Hopf algebra
ACM Communications in Computer Algebra
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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 polylogarithmsLi"s(z) and to special numbers, the multiple harmonic sumsH"s(N). In the ''good'' cases, both objects converge (respectively, as z-1 and as N-+~) to the same limit, the polyzeta@z(s). For the divergent cases, using the technologies of noncommutative generating series, we establish, by techniques ''a la Hopf'', a theorem ''a l'Abel'', involving the generating series of polyzetas. 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 H"s(N) 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.