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A tale of three structures: the arithmetics of multizetas, the analysis of singularities, the Lie algebra ARI. (English) Zbl 1065.11069

Braaksma, B. L. J. (ed.) et al., Differential equations and the Stokes phenomenon. Proceedings of the conference, Groningen, Netherlands, May 28–30, 2001. Singapore: World Scientific (ISBN 981-238-172-4/hbk). 89-146 (2002).
Summary: Two new, fast-developing, but at first sight completely disconnected subjects have turned out to be governed by a common underlying structure. These two subjects are: the specific singularities, Stokes phenomena and resurgence patterns exhibited by singularly perturbed systems; and the phenomenon of dimorphy (existence of a double product) displayed not only by the so-called multizeta values but by a host of other basic transcendental constants. As for the unifying structure, it is the novel Lie algebra ARI which, together with its group GARI and a number of related constructions, is a fascinating object in its own right.
Contents: 1. Introduction; 2. Singular systems and equational resurgence; 3. Singularly perturbed systems and co-equational resurgence; 4. Dimorphic monomials and monics; 5. The overarching structure: ARI/GARI; 6. The arithmetics of multizetas; 7. Conclusion and further vistas.
For the entire collection see [Zbl 1007.00033].

MSC:

11M41 Other Dirichlet series and zeta functions
17B99 Lie algebras and Lie superalgebras
34M40 Stokes phenomena and connection problems (linear and nonlinear) for ordinary differential equations in the complex domain
34M60 Singular perturbation problems for ordinary differential equations in the complex domain (complex WKB, turning points, steepest descent)
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