Darboux evaluations for hypergeometric functions with the projective monodromy $$\hbox {PSL}(2,\mathbb {F}_7)$$ PSL (
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Darboux evaluations for hypergeometric functions with the projective monodromy PSL(2, F7 ) Raimundas Vidunas1 Received: 7 September 2018 / Accepted: 12 November 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Algebraic hypergeometric functions can be compactly expressed as radical functions on pull-back curves where the monodromy group is simpler, say, a finite cyclic group. These so-called Darboux evaluations have already been considered for algebraic 2 F1 functions. This article presents Darboux evaluations for the classical case of 3 F2 functions with the projective monodromy group PSL(2, F7 ). The pullback curves are of genus 0 (in the simplest case) or of genus 1. As an application of the genus 0 evaluations, appealing modular evaluations of the same 3 F2 -functions are derived. Keywords Algebraic hypergeometric functions · Pull-back transformations · Monodromy · Klein’s curve · Modular functions Mathematics Subject Classification Primary 34A05 · 33C20; Secondary 34M15 · 11F03
1 Introduction One way to obtain workable expressions for algebraic hypergeometric functions is to pull-back them to algebraic curves where the (finite) monodromy group would be simpler, say, a finite cyclic group [34]. For example, we have 1 1/4, 7/12 x (x + 4)3 3/4 4(2x − 1)3 = 1 + 1 x (1 − 2x) , 4/3 4 3 3/2 x (x + 2) 1 1/2, 5/6 = 1 + 2x 2 F1 3 2 2/3 (2x + 1) (1 − x)
2 F1
B 1
(1.1) (1.2)
Raimundas Vidunas [email protected] Institute of Applied Mathematics, Vilnius University, Vilnius, Lithuania
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R. Vidunas
around x = 0. Here the 2 F1 -functions have the tetrahedral group ∼ = A4 as the projective monodromy group (of the hypergeometric differential equation). The rational arguments of degree 4 reduce the monodromy to small cyclic groups, as evidenced by the radical (i.e., algebraic power) functions on the right-hand sides of these identities. If a Fuchsian differential equation E on the Riemann sphere CP1 has a finite monodromy group, then E can be transformed by a pull-back transformation with respect to an algebraic covering ϕ : B → CP1 to a Fuchsian equation on the curve B with a cyclic monodromy. This is a special case of a Darboux covering as defined in [34]. The transformed equation on the Darboux curve B has a basis of radical solutions (and hence, a completely reducible monodromy representation). Explicit expressions for solutions of E in terms of radical functions on B are called Darboux evaluations. In [34], all tetrahedral, octahedral and icosahedral Schwarz types [24] of algebraic 2 F1 -functions are exemplified by Darboux evaluations. The simplest Darboux curves for most icosahedral Schwarz types have genus 1 rather than 0. Algebraic generalized hypergeometric functions p Fp−1 are classified by Beukers and Heckman [2]. One particularly interesting case [15,30] is algebraic 3 F2 -functions such that the projective monodromy group (of their third order Fuchsian equations) is the simple group (1.3) = PSL(2, F7 ) ∼ = GL(3, F2 ) with 168 elements. This i
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