University of Illinois Urbana-Champaign

Modular equations and Ramanujan's cubic and quartic theories of theta functions

Yuttanan, Boonrod

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Yuttanan_Boonrod.pdf

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https://hdl.handle.net/2142/29552

Description

Title
Modular equations and Ramanujan's cubic and quartic theories of theta functions
Author(s)
Yuttanan, Boonrod
Issue Date
2012-02-01T00:55:05Z
Director of Research (if dissertation) or Advisor (if thesis)
Berndt, Bruce C.
Committee Member(s)
  • Stolarsky, Kenneth B.
  • Berndt, Bruce C.
  • Zaharescu, Alexandru
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Date of Ingest
2012-02-01T00:55:05Z
Keyword(s)
  • modular equations
  • theta-functions
  • cubic theta-functions
  • eta-functions
  • partitions
  • colored partitions
  • continued fraction
  • power series expansion
  • periodicity of sign of coefficients
  • infinite series
Abstract
"In this thesis, we prove several identities involving Ramanujan's general theta function. In Chapter 2, we give proofs for new Ramanujan type modular equations discovered by Somos and establish applications of some of them. In Chapter 3, we will give proofs for several Dedekind eta product identities which Somos discovered through computational searches and which Choi discovered in his work on basic bilateral hypergeometric series and mock theta functions. In Chapter 4, we derive new identities related to the Ramanujan-G\""{o}llnitz-Gordon continued fraction that are similar to those for the famous Rogers-Ramanujan continued fraction. We give a new proof of the 8-dissection of the Ramanujan-G\""{o}llnitz-Gordon continued fraction and also show that the signs of the coefficients of power series associated with this continued fraction are periodic with period 8. In Chapter 5, we prove several infinite series identities involving hyperbolic functions and hypergeometric functions by using the classical and quartic theories of theta functions. In Chapter 6, we study a new function called a quartic analogue of Jacobian theta functions. Finally, Chapter 7 is devoted to establishing new identities related to the Borweins' cubic theta functions and Ramanujan's general theta function. We also give equivalent combinatorial interpretations of such identities."
Graduation Semester
2011-12
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http://hdl.handle.net/2142/29552
Copyright and License Information
Copyright 2011 Boonrod Yuttanan

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