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

Yuttanan, Boonrod

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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

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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Copyright and License Information

Copyright 2011 Boonrod Yuttanan

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