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arXiv · 2201.01451

On the tau function of the hypergeometric equation

Marco Bertola·Dmitry Korotkin
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Abstract

The monodromy map for a rank-two system of differential equations with three Fuchsian singularities is classically solved by the Kummer formul\ae\ 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.

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Marco Bertola, Dmitry Korotkin. 2022-01-05. On the tau function of the hypergeometric equation. https://doi.org/10.1016/j.physd.2022.133381

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