Multiple zeta values and the WKB method

Zakrzewski, Michał; Żołądek, Henryk

dc.contributor.author	Zakrzewski, Michał
dc.contributor.author	Żołądek, Henryk
dc.contributor.editor	Krasiński, Tadeusz
dc.contributor.editor	Spodzieja, Stanisław
dc.date.accessioned	2017-12-13T11:02:44Z
dc.date.available	2017-12-13T11:02:44Z
dc.date.issued	2013
dc.identifier.citation	Zakrzewski M., Żołądek H, Multiple zeta values and the WKB, [in:] Krasiński T., Spodzieja S. (eds.), Analytic and Algebraic Geometry, Łódź University Press, Łódź 2013, s. 155-202, doi: 10.18778/7969-017-6.12	pl_PL
dc.identifier.isbn	978-83-7969-017-6
dc.identifier.uri	http://hdl.handle.net/11089/23613
dc.description.abstract	The multiple zeta values ζ(d1, . . . , dr ) are natural generalizations of the values ζ(d) of the Riemann zeta functions at integers d. They have many applications, e.g. in knot theory and in quantum physics. It turns out that some generating functions for the multiple zeta values, like fd(x) = 1 − ζ(d)xd + ζ(d, d)x2d − . . . , are related with hypergeometric equations. More precisely, fd(x) is the value at t = 1 of some hypergeometric series dFd−1(t) = 1 − x t + . . ., a solution to a hypergeometric equation of degree d with parameter x. Our idea is to represent fd(x) as some connection coeffi- cient between certain standard bases of solutions near t = 0 and near t = 1. Moreover, we assume that \|x\| is large. For large complex x the above basic solutions are represented in terms of so-called WKB solutions. The series which define the WKB solutions are divergent and are subject to so-called Stokes phenomenon. Anyway it is possible to treat them rigorously. In the paper we review our results about application of the WKB method to the generating functions f x), focusing on the cases d = 2 and d = 3.	pl_PL
dc.language.iso	en	pl_PL
dc.publisher	Wydawnictwo Uniwersytetu Łódzkiego	pl_PL
dc.relation.ispartof	Krasiński T., Spodzieja S. (eds.), Analytic and Algebraic Geometry, Łódź University Press, Łódź 2013;
dc.rights	Uznanie autorstwa-Użycie niekomercyjne-Bez utworów zależnych 3.0 Polska	*
dc.rights.uri	http://creativecommons.org/licenses/by-nc-nd/3.0/pl/	*
dc.title	Multiple zeta values and the WKB method	pl_PL
dc.type	Book chapter	pl_PL
dc.rights.holder	© Copyright by University of Łódź, Łódź 2013	pl_PL
dc.page.number	155-202	pl_PL
dc.contributor.authorAffiliation	Institute of Mathematics, Jan Kochanowski University, ul. Świętokrzyska 15, 25-406 Kielce	pl_PL
dc.contributor.authorAffiliation	Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw	pl_PL
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dc.contributor.authorEmail	zakrzewski@mimuw.edu.pl	pl_PL
dc.contributor.authorEmail	zoladek@mimuw.edu.pl	pl_PL
dc.identifier.doi	10.18778/7969-017-6.12