Acta Mathematica Sinica, English Series Acta Mathematica Sinica, Published online: May 7, 2018 https://doi.org/10.1007/s10114-018-6389-2 English Series http://www.ActaMath.com Springer-Verlag GmbH Germany & The Editorial Office of AMS 2018 On Schwarzian Triangle Functions, Automorphic Forms and a Generalization of Ramanujan’s Triple Diﬀerential Equations Li Chien SHEN Department of Mathematics, University of Florida, Gainesville, FL 32611-8105,USA E-mail : shen@uﬂ.edu Dedicate to my teacher Professor Simon Hellerstein Abstract Let G be the group of the fractional linear transformations generated by π π τ cos +sin n n T (τ)= τ + λ, S(τ)= ; π π −τ sin +cos n n where π π cos +cos m n λ =2 ; sin m, n is a pair of integers with either n ≥ 2,m ≥ 3or n ≥ 3,m ≥ 2; τ lies in the upper half plane H. A fundamental set of functions f , f and f automorphic with respect to G will be constructed 0 i ∞ from the conformal mapping of the fundamental domain of G. We derive an analogue of Ramanujan’s triple diﬀerential equations associated with the group G and establish the connection of f , f and f 0 i ∞ with a family of hypergeometric functions.
Acta Mathematica Sinica, English Series – Springer Journals
Published: May 7, 2018
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