{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/29552"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/29552","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Modular equations and Ramanujan's cubic and quartic theories of theta functions","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.\"","abstract_html":""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."","abstract_has_math":false,"creators":["Yuttanan, Boonrod"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Berndt, Bruce C.","Stolarsky, Kenneth B.","Zaharescu, Alexandru"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2012,"date_issued":"2012-02-01T00:55:05Z","date_published":"2012-02-01T00:55:05Z","updated_at":"2026-07-22T22:25:27Z","subjects":["modular equations","theta-functions","cubic theta-functions","eta-functions","partitions","colored partitions","continued fraction","power series expansion","periodicity of sign of coefficients","infinite series"],"languages":["en"],"rights":["Copyright 2011 Boonrod Yuttanan"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/29552","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Berndt, Bruce C.","Stolarsky, Kenneth B.","Zaharescu, Alexandru"]},{"key":"dc:creator","label":"Author","values":["Yuttanan, Boonrod"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2012-02-01T00:55:05Z","2014-02-01T11:00:34Z","2011-12"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation / Thesis","text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["modular equations","theta-functions","cubic theta-functions","eta-functions","partitions","colored partitions","continued fraction","power series expansion","periodicity of sign of coefficients","infinite series"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2011 Boonrod Yuttanan"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/29552"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["\"In this thesis, we prove several identities involving Ramanujan's general theta function. 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