{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/23627"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/23627","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Contributions to Ramanujan's continued fractions, class invariants, partition identities and modular equations","abstract":"Various topics related to the work of Ramanujan are discussed in this thesis. In Chapter 2, we give a new proof of Ramanujan's famous partition identity modulo 5 (see (1.1)). This proof is an improvement of W. N. Bailey's proof given in 1952. We also establish a new proof of Ramanujan's partition identity modulo 7.","abstract_html":"Various topics related to the work of Ramanujan are discussed in this thesis. In Chapter 2, we give a new proof of Ramanujan's famous partition identity modulo 5 (see (1.1)). This proof is an improvement of W. N. Bailey's proof given in 1952. We also establish a new proof of Ramanujan's partition identity modulo 7.","abstract_has_math":false,"creators":["Chan, Heng Huat"],"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."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T14:21:07Z","date_published":"2011-05-07T14:21:07Z","updated_at":"2026-07-22T22:25:22Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1995 Chan, Heng Huat"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9543549","(UMI)AAI9543549"],"render_values":[{"text":"AAI9543549","href":null,"code":true},{"text":"(UMI)AAI9543549","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/23627","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Berndt, Bruce C."]},{"key":"dc:creator","label":"Author","values":["Chan, Heng Huat"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T14:21:07Z","10000-01-01","1995"]},{"key":"dc:type","label":"Dc Type","values":["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":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1995 Chan, Heng Huat"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9543549","(UMI)AAI9543549","http://hdl.handle.net/2142/23627"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Various topics related to the work of Ramanujan are discussed in this thesis. In Chapter 2, we give a new proof of Ramanujan's famous partition identity modulo 5 (see (1.1)). This proof is an improvement of W. N. Bailey's proof given in 1952. We also establish a new proof of Ramanujan's partition identity modulo 7.","One remarkable feature of Ramanujan's identities is that many of them appear in pairs. In Chapter 3, we explain this interesting phenomenon using Hecke's theory of correspondence between Fourier series and Dirichlet series.","Chapters 4 and 5 are devoted to the evaluations of Ramanujan-Weber class invariants. We establish 18 of these invariants which have not heretofore been proven. Our proofs rely heavily on the knowledge of modular equations and class field theory.","In Chapter 6, we study Ramanujan's cubic continued fraction G(q) (see (1.7)) and construct relations between various continued fractions. We also use the results of Chapter 4 to give explicit evaluations of G(q) at $q=\\pm e\\sp{-\\pi\\sqrt{n}}$.","Undoubtedly, one of Ramanujan's favorite topics is the Rogers-Ramanujan continued fraction F(q) (see (1.6)). In Chapter 7, using modular equations of degrees 5 and 25, we establish theorems which enable us to evaluate F(q) at $q=e\\sp{-2}\\pi\\sqrt{n}$ and $-e\\sp{-\\pi\\sqrt{n}}$. In particular, we are able to complete a table initiated by Ramanujan on page 210 of his Lost Notebook.","In his first notebook, Ramanujan recorded several values of the classical theta function $\\varphi(q)$ (see (2.1.7)). In our final chapter, we give natural proofs of these values using modular equations of various degrees. We also discover a new identity which is related to the Borweins' cubic theta functions.","Made available in DSpace on 2011-05-07T14:21:07Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9543549.pdf: 3235294 bytes, checksum: 539060ae00b9eda07640ecd7aa0a5c4b (MD5) Previous issue date: 1995","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T15:05:45Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:31:32-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"]},{"key":"dc:title","label":"Title","values":["Contributions to Ramanujan's continued fractions, class invariants, partition identities and modular equations"]}]}],"canonical_facts":{"dc:contributor":["Berndt, Bruce C."],"dc:creator":["Chan, Heng Huat"],"dc:date":["2011-05-07T14:21:07Z","10000-01-01","1995"],"dc:description":["Various topics related to the work of Ramanujan are discussed in this thesis. In Chapter 2, we give a new proof of Ramanujan's famous partition identity modulo 5 (see (1.1)). This proof is an improvement of W. N. Bailey's proof given in 1952. We also establish a new proof of Ramanujan's partition identity modulo 7.","One remarkable feature of Ramanujan's identities is that many of them appear in pairs. In Chapter 3, we explain this interesting phenomenon using Hecke's theory of correspondence between Fourier series and Dirichlet series.","Chapters 4 and 5 are devoted to the evaluations of Ramanujan-Weber class invariants. We establish 18 of these invariants which have not heretofore been proven. Our proofs rely heavily on the knowledge of modular equations and class field theory.","In Chapter 6, we study Ramanujan's cubic continued fraction G(q) (see (1.7)) and construct relations between various continued fractions. We also use the results of Chapter 4 to give explicit evaluations of G(q) at $q=\\pm e\\sp{-\\pi\\sqrt{n}}$.","Undoubtedly, one of Ramanujan's favorite topics is the Rogers-Ramanujan continued fraction F(q) (see (1.6)). In Chapter 7, using modular equations of degrees 5 and 25, we establish theorems which enable us to evaluate F(q) at $q=e\\sp{-2}\\pi\\sqrt{n}$ and $-e\\sp{-\\pi\\sqrt{n}}$. In particular, we are able to complete a table initiated by Ramanujan on page 210 of his Lost Notebook.","In his first notebook, Ramanujan recorded several values of the classical theta function $\\varphi(q)$ (see (2.1.7)). In our final chapter, we give natural proofs of these values using modular equations of various degrees. We also discover a new identity which is related to the Borweins' cubic theta functions.","Made available in DSpace on 2011-05-07T14:21:07Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9543549.pdf: 3235294 bytes, checksum: 539060ae00b9eda07640ecd7aa0a5c4b (MD5) Previous issue date: 1995","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T15:05:45Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:31:32-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"],"dc:identifier":["AAI9543549","(UMI)AAI9543549","http://hdl.handle.net/2142/23627"],"dc:language":["eng"],"dc:rights":["Copyright 1995 Chan, Heng Huat"],"dc:subject":["Mathematics"],"dc:title":["Contributions to Ramanujan's continued fractions, class invariants, partition identities and modular equations"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:22Z"}