{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/86824"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/86824","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Contributions to Trigonometric Sums and Fifth -Order Mock Theta Functions","abstract":"\"This thesis explores two different topics in number theory. The first portion of this thesis concentrates on trigonometric sums, giving also a survey on the history of evaluations and reciprocity theorems on such sums. We establish some general theorems on explicit evaluations and reciprocity theorems using contour integration as our main method. The second portion of this thesis focuses on Ramanujan's mock theta functions. In the second chapter, we extend the work of G. N. Watson on the fifth order mock theta functions, and show that in comparison to the third order functions, the fifth order mock theta functions possess a more complex behavior on radial approach to rational points on the unit circle. We prove a special case of a general transformation formula for the fifth order mock theta functions, extending the work of G. E. Andrews on the third order functions. Finally, the last chapter is devoted to proving four mock theta identities found in Ramanujan's \"\"lost\"\" notebook. We devise a method based on Ramanujan's y11 summation formula which effectively allows the computation of the residues of all the simple poles of certain infinite products. We explain how the method may be applied to obtain new identities of a similar type. The main ingredient in our proofs is a lemma of A. O. L. Atkin and P. Swinnerton-Dyer.\"","abstract_html":""This thesis explores two different topics in number theory. The first portion of this thesis concentrates on trigonometric sums, giving also a survey on the history of evaluations and reciprocity theorems on such sums. We establish some general theorems on explicit evaluations and reciprocity theorems using contour integration as our main method. The second portion of this thesis focuses on Ramanujan's mock theta functions. In the second chapter, we extend the work of G. N. Watson on the fifth order mock theta functions, and show that in comparison to the third order functions, the fifth order mock theta functions possess a more complex behavior on radial approach to rational points on the unit circle. We prove a special case of a general transformation formula for the fifth order mock theta functions, extending the work of G. E. Andrews on the third order functions. Finally, the last chapter is devoted to proving four mock theta identities found in Ramanujan's ""lost"" notebook. We devise a method based on Ramanujan's y11 summation formula which effectively allows the computation of the residues of all the simple poles of certain infinite products. We explain how the method may be applied to obtain new identities of a similar type. The main ingredient in our proofs is a lemma of A. O. L. Atkin and P. Swinnerton-Dyer."","abstract_has_math":false,"creators":["Yeap, Boon Pin"],"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":2015,"date_issued":"2015-09-28T15:19:43Z","date_published":"2015-09-28T15:19:43Z","updated_at":"2026-07-22T22:26:28Z","subjects":["Mathematics"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(MiAaPQ)AAI3102001"],"render_values":[{"text":"(MiAaPQ)AAI3102001","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/86824","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":["Yeap, Boon Pin"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2015-09-28T15:19:43Z","10000-01-01","2003"]},{"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"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/86824","(MiAaPQ)AAI3102001"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["\"This thesis explores two different topics in number theory. The first portion of this thesis concentrates on trigonometric sums, giving also a survey on the history of evaluations and reciprocity theorems on such sums. We establish some general theorems on explicit evaluations and reciprocity theorems using contour integration as our main method. The second portion of this thesis focuses on Ramanujan's mock theta functions. In the second chapter, we extend the work of G. N. Watson on the fifth order mock theta functions, and show that in comparison to the third order functions, the fifth order mock theta functions possess a more complex behavior on radial approach to rational points on the unit circle. We prove a special case of a general transformation formula for the fifth order mock theta functions, extending the work of G. E. Andrews on the third order functions. Finally, the last chapter is devoted to proving four mock theta identities found in Ramanujan's \"\"lost\"\" notebook. We devise a method based on Ramanujan's y11 summation formula which effectively allows the computation of the residues of all the simple poles of certain infinite products. We explain how the method may be applied to obtain new identities of a similar type. The main ingredient in our proofs is a lemma of A. O. L. Atkin and P. Swinnerton-Dyer.\"","Made available in DSpace on 2015-09-28T15:19:43Z (GMT). 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The first portion of this thesis concentrates on trigonometric sums, giving also a survey on the history of evaluations and reciprocity theorems on such sums. We establish some general theorems on explicit evaluations and reciprocity theorems using contour integration as our main method. The second portion of this thesis focuses on Ramanujan's mock theta functions. In the second chapter, we extend the work of G. N. Watson on the fifth order mock theta functions, and show that in comparison to the third order functions, the fifth order mock theta functions possess a more complex behavior on radial approach to rational points on the unit circle. We prove a special case of a general transformation formula for the fifth order mock theta functions, extending the work of G. E. Andrews on the third order functions. Finally, the last chapter is devoted to proving four mock theta identities found in Ramanujan's \"\"lost\"\" notebook. We devise a method based on Ramanujan's y11 summation formula which effectively allows the computation of the residues of all the simple poles of certain infinite products. We explain how the method may be applied to obtain new identities of a similar type. The main ingredient in our proofs is a lemma of A. O. L. Atkin and P. Swinnerton-Dyer.\"","Made available in DSpace on 2015-09-28T15:19:43Z (GMT). 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