Noncommutative algebra, multiple harmonic sums and applications in discrete probability

• Autores: Christian Costermans, Hoang Ngoc Minh
• Localización: Journal of symbolic computation, ISSN 0747-7171, Vol. 44, Nº 7, 2009, págs. 801-817
• Idioma: inglés
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• Resumen

  • 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?+8) 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?+8. 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.