This article is cited in 9 papers

Fuchsian Equations with Three Non-Apparent Singularities

Alexandre Eremenko^a, Vitaly Tarasov<sup>bc

a</sup> Purdue University, West Lafayette, IN 47907, USA
^b St. Petersburg Branch of Steklov Mathematical Institute, St. Petersburg, 191023, Russia
^c Indiana University – Purdue University Indianapolis, Indianapolis, IN 46202, USA

Abstract: We show that for every second order Fuchsian linear differential equation $E$ with $n$ singularities of which $n-3$ are apparent there exists a hypergeometric equation $H$ and a linear differential operator with polynomial coefficients which maps the space of solutions of $H$ into the space of solutions of $E$. This map is surjective for generic parameters. This justifies one statement of Klein (1905). We also count the number of such equations $E$ with prescribed singularities and exponents. We apply these results to the description of conformal metrics of curvature $1$ on the punctured sphere with conic singularities, all but three of them having integer angles.

Keywords: Fuchsian equations; hypergeometric equation; difference equations; apparent singularities; bispectral duality; positive curvature; conic singularities.

MSC: 34M03; 34M35; 57M50

Received: February 2, 2018; in final form June 10, 2018; Published online June 15, 2018

Language: English

DOI: 10.3842/SIGMA.2018.058

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