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On Schwarzian Triangle Functions, Automorphic Forms and a Generalization of Ramanujan’s Triple Differential Equations

On Schwarzian Triangle Functions, Automorphic Forms and a Generalization of Ramanujan’s Triple... Let G be the group of the fractional linear transformations generated by $$T(\tau ) = \tau + \lambda ,S(\tau ) = \frac{{\tau \cos \frac{\pi }{n} + \sin \frac{\pi }{n}}}{{ - \tau \sin \frac{\pi }{n} + \cos \frac{\pi }{n}}};$$ T ( τ ) = τ + λ , S ( τ ) = τ cos π n + sin π n − τ sin π n + cos π n ; where $$\lambda = 2\frac{{\cos \frac{\pi }{m} + \cos \frac{\pi }{n}}}{{\sin \frac{\pi }{n}}};$$ λ = 2 cos π m + cos π n sin π n ; m, n is a pair of integers with either n ≥ 2,m ≥ 3 or n ≥ 3,m ≥ 2; τ lies in the upper half plane H. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematica Sinica, English Series Springer Journals

On Schwarzian Triangle Functions, Automorphic Forms and a Generalization of Ramanujan’s Triple Differential Equations

Shen, Li
Acta Mathematica Sinica, English Series , Volume 34 (11) – May 7, 2018
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Publisher
Springer Journals
Copyright
Copyright © 2018 by Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature
Subject
Mathematics; Mathematics, general
ISSN
1439-8516
eISSN
1439-7617
DOI
10.1007/s10114-018-6389-2
Publisher site
See Article on Publisher Site

Abstract

Let G be the group of the fractional linear transformations generated by $$T(\tau ) = \tau + \lambda ,S(\tau ) = \frac{{\tau \cos \frac{\pi }{n} + \sin \frac{\pi }{n}}}{{ - \tau \sin \frac{\pi }{n} + \cos \frac{\pi }{n}}};$$ T ( τ ) = τ + λ , S ( τ ) = τ cos π n + sin π n − τ sin π n + cos π n ; where $$\lambda = 2\frac{{\cos \frac{\pi }{m} + \cos \frac{\pi }{n}}}{{\sin \frac{\pi }{n}}};$$ λ = 2 cos π m + cos π n sin π n ; m, n is a pair of integers with either n ≥ 2,m ≥ 3 or n ≥ 3,m ≥ 2; τ lies in the upper half plane H.

Journal

Acta Mathematica Sinica, English Series – Springer Journals

Published: May 7, 2018

References

  • (Editor): Higher Transcendental Functions
    Erdélyi, A.
  • Ramanujan, Chelsea
    Hardy, G. H.
  • Lectures on Dirichlet Series, Modular Functions and Quadratic Forms
    Hecke, E.
  • Conformal Mappings
    Nehari, Z.
  • A note on the Schwarzian triangle functions
    Shen, L. C.
  • On the logarithmic derivative of a theta function and a fundamental identity of Ramanujan
    Shen, L. C.
  • Hecke groups, Schwarzian triangle functions and a class of hyper-elliptic functions
    Shen, L. C.
  • Elliptic Functions According to Eisenstein and Kronecker
    Weil, A.
  • A Course of Modern Analysis
    Whittaker, E. T.; Watson, G. N.