University of Illinois Urbana-Champaign

Contributions to Ramanujan's continued fractions, class invariants, partition identities and modular equations

Chan, Heng Huat

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Description

Title
Contributions to Ramanujan's continued fractions, class invariants, partition identities and modular equations
Author(s)
Chan, Heng Huat
Issue Date
1995
Doctoral Committee Chair(s)
Berndt, Bruce C.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
  • Various topics related to the work of Ramanujan are discussed in this thesis. In Chapter 2, we give a new proof of Ramanujan's famous partition identity modulo 5 (see (1.1)). This proof is an improvement of W. N. Bailey's proof given in 1952. We also establish a new proof of Ramanujan's partition identity modulo 7.
  • One remarkable feature of Ramanujan's identities is that many of them appear in pairs. In Chapter 3, we explain this interesting phenomenon using Hecke's theory of correspondence between Fourier series and Dirichlet series.
  • Chapters 4 and 5 are devoted to the evaluations of Ramanujan-Weber class invariants. We establish 18 of these invariants which have not heretofore been proven. Our proofs rely heavily on the knowledge of modular equations and class field theory.
  • In Chapter 6, we study Ramanujan's cubic continued fraction G(q) (see (1.7)) and construct relations between various continued fractions. We also use the results of Chapter 4 to give explicit evaluations of G(q) at $q=\pm e\sp{-\pi\sqrt{n}}$.
  • Undoubtedly, one of Ramanujan's favorite topics is the Rogers-Ramanujan continued fraction F(q) (see (1.6)). In Chapter 7, using modular equations of degrees 5 and 25, we establish theorems which enable us to evaluate F(q) at $q=e\sp{-2}\pi\sqrt{n}$ and $-e\sp{-\pi\sqrt{n}}$. In particular, we are able to complete a table initiated by Ramanujan on page 210 of his Lost Notebook.
  • In his first notebook, Ramanujan recorded several values of the classical theta function $\varphi(q)$ (see (2.1.7)). In our final chapter, we give natural proofs of these values using modular equations of various degrees. We also discover a new identity which is related to the Borweins' cubic theta functions.
Type of Resource
text
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http://hdl.handle.net/2142/23627
Copyright and License Information
Copyright 1995 Chan, Heng Huat

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