{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/18485"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/18485","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Modular identities for the Rogers-Ramanujan functions and analogues","abstract":"Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by William Ingram (wingram2@illinois.edu) on 2011-01-21T22:47:37Z Item is restricted until 2013-01-21T22:47:37Z","abstract_html":"Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by William Ingram (wingram2@illinois.edu) on 2011-01-21T22:47:37Z Item is restricted until 2013-01-21T22:47:37Z","abstract_has_math":false,"creators":["Gugg, Chadwick"],"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.","Ahlgren, Scott","Ford, Kevin","Stolarsky, Kenneth B."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-01-21T22:42:26Z","date_published":"2011-01-21T22:42:26Z","updated_at":"2026-07-22T22:25:11Z","subjects":["Rogers-Ramanujan functions","Rogers-Ramanujan continued fraction","Ramanujan's notebooks","Ramanujan's Lost Notebook","theta functions","modular relations","partitions","q-series","Gollnitz-Gordon functions","Gollnitz-Gordon continued fraction"],"languages":["en"],"rights":["Copyright 2010 Chadwick Gugg"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/18485","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Berndt, Bruce C.","Ahlgren, Scott","Ford, Kevin","Stolarsky, Kenneth B."]},{"key":"dc:creator","label":"Author","values":["Gugg, Chadwick"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-01-21T22:42:26Z","2013-01-22T11:00:23Z","2010-12"]},{"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":["Rogers-Ramanujan functions","Rogers-Ramanujan continued fraction","Ramanujan's notebooks","Ramanujan's Lost Notebook","theta functions","modular relations","partitions","q-series","Gollnitz-Gordon functions","Gollnitz-Gordon continued fraction"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2010 Chadwick Gugg"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/18485"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by William Ingram (wingram2@illinois.edu) on 2011-01-21T22:47:37Z Item is restricted until 2013-01-21T22:47:37Z","\"In his notebooks, Ramanujan recorded 40 beautiful modular relations for the Rogers-Ramanujan functions. Of these 40 identities, precisely one involves powers of the Rogers-Ramanujan functions. Ramanujan added the enigmatic note that \"\"Each of these formulae is the simplest of a large class.\"\" This suggests that there are further modular identities involving powers of the Rogers-Ramanujan functions. Although numerous authors have studied identities for the Rogers-Ramanujan functions and various analogues, no systematic study of identities involving powers of the Rogers-Ramanujan functions has been undertaken. In this thesis, we continue the study of modular identities for the Rogers-Ramanujan functions, with particular emphasis on relations involving powers of the Rogers-Ramanujan functions. Our methods are classical, using tools and techniques that Ramanujan could have employed. These tools include, for example, manipulation of infinite series and the theory of modular equations. It is hoped that these methods will give new insights into these equations, and perhaps lead to understanding or discovering further families of identities of mathematical interest. Identities involving squares, cubes, fourth, and fifth powers of the Rogers-Ramanujan functions are enunciated and proved; many of these relations are new. Rich applications are made to the study of modular relations for the Rogers-Ramanujan continued fraction. To demonstrate the generality of our methods, analogous results are obtained in various cases for the Gollnitz-Gordon functions and the Ramanujan-Gollnitz-Gordon continued fraction. Further identities for the Rogers-Ramanujan functions, of the types found in Ramanujan's list of 40 relations for the Rogers-Ramanujan functions, are also studied. Analogous identities are obtained for the Gollnitz-Gordon functions, as well as for dodecic and sextodecic analogues of the Rogers-Ramanujan functions.\"","Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2010-06-23T13:20:26Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 2 Gugg_Chadwick.tex: 432595 bytes, checksum: 3941dc3589a2127aa70541b73b58bf17 (MD5) Gugg_Chadwick.pdf: 602058 bytes, checksum: fda94a76b8df5118b431c615468b75cc (MD5)","Made available in DSpace on 2011-01-21T22:42:26Z (GMT). No. of bitstreams: 3 Gugg_Chadwick.pdf: 602058 bytes, checksum: fda94a76b8df5118b431c615468b75cc (MD5) license.txt: 4066 bytes, checksum: 301f7d682fbd12f7bf6fc08fe360fc28 (MD5) Gugg_Chadwick.tex: 432595 bytes, checksum: 3941dc3589a2127aa70541b73b58bf17 (MD5)","Item reinstated by Sarah Shreeves (sshreeve@illinois.edu) on 2013-01-22T11:00:23Z Item was in collections: University of Illinois Dissertations and Theses (ID: 204) Dissertations and Theses - Mathematics (ID: 749) No. of bitstreams: 4 Gugg_Chadwick.pdf: 602058 bytes, checksum: fda94a76b8df5118b431c615468b75cc (MD5) license.txt: 4066 bytes, checksum: 301f7d682fbd12f7bf6fc08fe360fc28 (MD5) Gugg_Chadwick.tex: 432595 bytes, checksum: 3941dc3589a2127aa70541b73b58bf17 (MD5) Gugg_Chadwick.pdf.txt: 239207 bytes, checksum: 12cb3a31a8b38f2b5136baf1a36784f9 (MD5)","Item released from any restrictions by Sarah Shreeves (sshreeve@illinois.edu) on 2013-01-22T11:00:23Z"]},{"key":"dc:title","label":"Title","values":["Modular identities for the Rogers-Ramanujan functions and analogues"]}]}],"canonical_facts":{"dc:contributor":["Berndt, Bruce C.","Ahlgren, Scott","Ford, Kevin","Stolarsky, Kenneth B."],"dc:creator":["Gugg, Chadwick"],"dc:date":["2011-01-21T22:42:26Z","2013-01-22T11:00:23Z","2010-12"],"dc:description":["Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by William Ingram (wingram2@illinois.edu) on 2011-01-21T22:47:37Z Item is restricted until 2013-01-21T22:47:37Z","\"In his notebooks, Ramanujan recorded 40 beautiful modular relations for the Rogers-Ramanujan functions. Of these 40 identities, precisely one involves powers of the Rogers-Ramanujan functions. Ramanujan added the enigmatic note that \"\"Each of these formulae is the simplest of a large class.\"\" This suggests that there are further modular identities involving powers of the Rogers-Ramanujan functions. Although numerous authors have studied identities for the Rogers-Ramanujan functions and various analogues, no systematic study of identities involving powers of the Rogers-Ramanujan functions has been undertaken. In this thesis, we continue the study of modular identities for the Rogers-Ramanujan functions, with particular emphasis on relations involving powers of the Rogers-Ramanujan functions. Our methods are classical, using tools and techniques that Ramanujan could have employed. These tools include, for example, manipulation of infinite series and the theory of modular equations. It is hoped that these methods will give new insights into these equations, and perhaps lead to understanding or discovering further families of identities of mathematical interest. Identities involving squares, cubes, fourth, and fifth powers of the Rogers-Ramanujan functions are enunciated and proved; many of these relations are new. Rich applications are made to the study of modular relations for the Rogers-Ramanujan continued fraction. To demonstrate the generality of our methods, analogous results are obtained in various cases for the Gollnitz-Gordon functions and the Ramanujan-Gollnitz-Gordon continued fraction. Further identities for the Rogers-Ramanujan functions, of the types found in Ramanujan's list of 40 relations for the Rogers-Ramanujan functions, are also studied. Analogous identities are obtained for the Gollnitz-Gordon functions, as well as for dodecic and sextodecic analogues of the Rogers-Ramanujan functions.\"","Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2010-06-23T13:20:26Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 2 Gugg_Chadwick.tex: 432595 bytes, checksum: 3941dc3589a2127aa70541b73b58bf17 (MD5) Gugg_Chadwick.pdf: 602058 bytes, checksum: fda94a76b8df5118b431c615468b75cc (MD5)","Made available in DSpace on 2011-01-21T22:42:26Z (GMT). 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