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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 UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  modular equations
thetafunctions cubic thetafunctions etafunctions 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 RamanujanG\"{o}llnitzGordon continued fraction that are similar to those for the famous RogersRamanujan continued fraction. We give a new proof of the 8dissection of the RamanujanG\"{o}llnitzGordon 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:  20120201 
Genre:  thesis 
URI:  http://hdl.handle.net/2142/29552 
Rights Information:  Copyright 2011 Boonrod Yuttanan 
Date Available in IDEALS:  20140201 
Date Deposited:  201112 
This item appears in the following Collection(s)

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois 
Dissertations and Theses  Mathematics