On the tau function of the hypergeometric equation

DOI10.1016/J.PHYSD.2022.133381arXiv2201.01451WikidataQ113866778 ScholiaQ113866778MaRDI QIDQ6387527

D. Korotkin, M. Bertola

Publication date: 4 January 2022

Abstract: The monodromy map for a rank-two system of differential equations with three Fuchsian singularities is classically solved by the Kummer formulae for Gauss' hypergeometric functions. We define the tau-function of such a system as the generating function of the extended monodromy symplectomorphism, using an idea recently developed. This formulation allows us to determine the dependence of the tau-function on the monodromy data. Using the explicit solution of the monodromy problem, the tau-function is then explicitly written in terms of Barnes $G$-function. In particular, if the Fuchsian singularities are placed to $0$, $1$ and $infty$, this gives the structure constants of the asymptotical formula of Iorgov-Gamayun-Lisovyy for solutions of Painlev'e VI equation.

Mathematics Subject Classification ID

Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies (34M55) Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms (34M35) Isomonodromic deformations for ordinary differential equations in the complex domain (34M56)

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