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Title:Modular equations and Ramanujan's cubic and quartic theories of theta functions
Author(s):Yuttanan, Boonrod
Director of Research:Berndt, Bruce C.
Doctoral Committee Member(s):Stolarsky, Kenneth B.; Berndt, Bruce C.; Zaharescu, Alexandru
Department / Program:Mathematics
Discipline:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Degree:Ph.D.
Genre:Dissertation
Subject(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.
Issue Date:2012-02-01
Genre:thesis
URI:http://hdl.handle.net/2142/29552
Rights Information:Copyright 2011 Boonrod Yuttanan
Date Available in IDEALS:2014-02-01
Date Deposited:2011-12


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