Unimodal sequences and quantum and mock modular forms
We show that the rank generating function U (t ; q) for strongly unimodal sequences lies at the interface of quantum modular forms and mock modular forms. We use U (-1; q) to obtain a quantum modular form which is “dual” to the quantum form Zagier constructed from Kontsevich’s “strange” function F (...
Gespeichert in:
| Veröffentlicht in: | Proceedings of the National Academy of Sciences - PNAS 2012-10, Vol.109 (40), p.16063-16067 |
|---|---|
| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | eng |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| container_end_page | 16067 |
|---|---|
| container_issue | 40 |
| container_start_page | 16063 |
| container_title | Proceedings of the National Academy of Sciences - PNAS |
| container_volume | 109 |
| creator | Bryson, Jennifer Ono, Ken Pitman, Sarah Rhoades, Robert C |
| description | We show that the rank generating function U (t ; q) for strongly unimodal sequences lies at the interface of quantum modular forms and mock modular forms. We use U (-1; q) to obtain a quantum modular form which is “dual” to the quantum form Zagier constructed from Kontsevich’s “strange” function F (q). As a result, we obtain a new representation for a certain generating function for L -values. The series U (i ; q) = U (- i ; q) is a mock modular form, and we use this fact to obtain new congruences for certain enumerative functions. |
| doi_str_mv | 10.1073/pnas.1211964109 |
| format | Article |
| fullrecord | <record><control><sourceid>jstor_pnas_</sourceid><recordid>TN_cdi_pnas_primary_109_40_16063</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><jstor_id>41763202</jstor_id><sourcerecordid>41763202</sourcerecordid><originalsourceid>FETCH-LOGICAL-c507t-830332930bcb61d69c94b8b4c518dcd0b8b4be669edd90c6fac6315914c4a3413</originalsourceid><addsrcrecordid>eNpVUEtLxDAQDqLgunr2JPbopetMk6bNRRDxBYIH3XNIk1SrbbMmreC_N7WL4mVm4HsxHyHHCCuEgp5vehVWmCEKzhDEDlnEiSlnAnbJAiAr0pJlbJ8chPAGACIvYUHydd90zqg2CfZjtL22IVG9ST5G1Q9j93N3Tr_HYcZW-aR2vguHZK9WbbBH270k65vr56u79OHx9v7q8iHVORRDWlKgNBMUKl1xNFxowaqyYjrH0mgD011ZzoU1RoDmtdKcYi6QaaYoQ7okF7PvZqw6a7TtB69aufFNp_yXdKqR_5G-eZUv7lNSVoic02hwtjXwLr4XBtk1Qdu2Vb11Y5BYAkWWsyKL1POZqr0Lwdv6NwZBTg3LqWH513BUJFvzCfhjC8migsNP_slMeQuD878chgWnGUyhpzNeKyfVi2-CXD9lEMWAmShzSr8B9wiM3A</addsrcrecordid><sourcetype>Open Access Repository</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>1803145472</pqid></control><display><type>article</type><title>Unimodal sequences and quantum and mock modular forms</title><source>Jstor Complete Legacy</source><source>PubMed Central</source><source>Alma/SFX Local Collection</source><source>Free Full-Text Journals in Chemistry</source><creator>Bryson, Jennifer ; Ono, Ken ; Pitman, Sarah ; Rhoades, Robert C</creator><creatorcontrib>Bryson, Jennifer ; Ono, Ken ; Pitman, Sarah ; Rhoades, Robert C</creatorcontrib><description>We show that the rank generating function U (t ; q) for strongly unimodal sequences lies at the interface of quantum modular forms and mock modular forms. We use U (-1; q) to obtain a quantum modular form which is “dual” to the quantum form Zagier constructed from Kontsevich’s “strange” function F (q). As a result, we obtain a new representation for a certain generating function for L -values. The series U (i ; q) = U (- i ; q) is a mock modular form, and we use this fact to obtain new congruences for certain enumerative functions.</description><identifier>ISSN: 0027-8424</identifier><identifier>EISSN: 1091-6490</identifier><identifier>DOI: 10.1073/pnas.1211964109</identifier><language>eng</language><publisher>National Academy of Sciences</publisher><subject>College mathematics ; equations ; functional response models ; Generating function ; Integers ; Mathematical congruence ; Mathematical cusps ; Mathematical functions ; Mathematical sequences ; Mathematical theorems ; Physical Sciences ; Series convergence ; statistical analysis</subject><ispartof>Proceedings of the National Academy of Sciences - PNAS, 2012-10, Vol.109 (40), p.16063-16067</ispartof><rights>copyright © 1993-2008 National Academy of Sciences of the United States of America</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c507t-830332930bcb61d69c94b8b4c518dcd0b8b4be669edd90c6fac6315914c4a3413</citedby><cites>FETCH-LOGICAL-c507t-830332930bcb61d69c94b8b4c518dcd0b8b4be669edd90c6fac6315914c4a3413</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Uhttp://www.pnas.org/content/109/40.cover.gif</thumbnail><linktopdf>$$Uhttps://www.jstor.org/stable/pdf/41763202$$EPDF$$P50$$Gjstor$$H</linktopdf><linktohtml>$$Uhttps://www.jstor.org/stable/41763202$$EHTML$$P50$$Gjstor$$H</linktohtml><link.rule.ids>230,314,723,776,780,799,881,27903,27904,53769,53771,57995,58228</link.rule.ids></links><search><creatorcontrib>Bryson, Jennifer</creatorcontrib><creatorcontrib>Ono, Ken</creatorcontrib><creatorcontrib>Pitman, Sarah</creatorcontrib><creatorcontrib>Rhoades, Robert C</creatorcontrib><title>Unimodal sequences and quantum and mock modular forms</title><title>Proceedings of the National Academy of Sciences - PNAS</title><description>We show that the rank generating function U (t ; q) for strongly unimodal sequences lies at the interface of quantum modular forms and mock modular forms. We use U (-1; q) to obtain a quantum modular form which is “dual” to the quantum form Zagier constructed from Kontsevich’s “strange” function F (q). As a result, we obtain a new representation for a certain generating function for L -values. The series U (i ; q) = U (- i ; q) is a mock modular form, and we use this fact to obtain new congruences for certain enumerative functions.</description><subject>College mathematics</subject><subject>equations</subject><subject>functional response models</subject><subject>Generating function</subject><subject>Integers</subject><subject>Mathematical congruence</subject><subject>Mathematical cusps</subject><subject>Mathematical functions</subject><subject>Mathematical sequences</subject><subject>Mathematical theorems</subject><subject>Physical Sciences</subject><subject>Series convergence</subject><subject>statistical analysis</subject><issn>0027-8424</issn><issn>1091-6490</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2012</creationdate><recordtype>article</recordtype><recordid>eNpVUEtLxDAQDqLgunr2JPbopetMk6bNRRDxBYIH3XNIk1SrbbMmreC_N7WL4mVm4HsxHyHHCCuEgp5vehVWmCEKzhDEDlnEiSlnAnbJAiAr0pJlbJ8chPAGACIvYUHydd90zqg2CfZjtL22IVG9ST5G1Q9j93N3Tr_HYcZW-aR2vguHZK9WbbBH270k65vr56u79OHx9v7q8iHVORRDWlKgNBMUKl1xNFxowaqyYjrH0mgD011ZzoU1RoDmtdKcYi6QaaYoQ7okF7PvZqw6a7TtB69aufFNp_yXdKqR_5G-eZUv7lNSVoic02hwtjXwLr4XBtk1Qdu2Vb11Y5BYAkWWsyKL1POZqr0Lwdv6NwZBTg3LqWH513BUJFvzCfhjC8migsNP_slMeQuD878chgWnGUyhpzNeKyfVi2-CXD9lEMWAmShzSr8B9wiM3A</recordid><startdate>20121002</startdate><enddate>20121002</enddate><creator>Bryson, Jennifer</creator><creator>Ono, Ken</creator><creator>Pitman, Sarah</creator><creator>Rhoades, Robert C</creator><general>National Academy of Sciences</general><general>National Acad Sciences</general><scope>FBQ</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>7S9</scope><scope>L.6</scope><scope>5PM</scope></search><sort><creationdate>20121002</creationdate><title>Unimodal sequences and quantum and mock modular forms</title><author>Bryson, Jennifer ; Ono, Ken ; Pitman, Sarah ; Rhoades, Robert C</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c507t-830332930bcb61d69c94b8b4c518dcd0b8b4be669edd90c6fac6315914c4a3413</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2012</creationdate><topic>College mathematics</topic><topic>equations</topic><topic>functional response models</topic><topic>Generating function</topic><topic>Integers</topic><topic>Mathematical congruence</topic><topic>Mathematical cusps</topic><topic>Mathematical functions</topic><topic>Mathematical sequences</topic><topic>Mathematical theorems</topic><topic>Physical Sciences</topic><topic>Series convergence</topic><topic>statistical analysis</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Bryson, Jennifer</creatorcontrib><creatorcontrib>Ono, Ken</creatorcontrib><creatorcontrib>Pitman, Sarah</creatorcontrib><creatorcontrib>Rhoades, Robert C</creatorcontrib><collection>AGRIS</collection><collection>CrossRef</collection><collection>AGRICOLA</collection><collection>AGRICOLA - Academic</collection><collection>PubMed Central (Full Participant titles)</collection><jtitle>Proceedings of the National Academy of Sciences - PNAS</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Bryson, Jennifer</au><au>Ono, Ken</au><au>Pitman, Sarah</au><au>Rhoades, Robert C</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Unimodal sequences and quantum and mock modular forms</atitle><jtitle>Proceedings of the National Academy of Sciences - PNAS</jtitle><date>2012-10-02</date><risdate>2012</risdate><volume>109</volume><issue>40</issue><spage>16063</spage><epage>16067</epage><pages>16063-16067</pages><issn>0027-8424</issn><eissn>1091-6490</eissn><abstract>We show that the rank generating function U (t ; q) for strongly unimodal sequences lies at the interface of quantum modular forms and mock modular forms. We use U (-1; q) to obtain a quantum modular form which is “dual” to the quantum form Zagier constructed from Kontsevich’s “strange” function F (q). As a result, we obtain a new representation for a certain generating function for L -values. The series U (i ; q) = U (- i ; q) is a mock modular form, and we use this fact to obtain new congruences for certain enumerative functions.</abstract><pub>National Academy of Sciences</pub><doi>10.1073/pnas.1211964109</doi><tpages>5</tpages><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0027-8424 |
| ispartof | Proceedings of the National Academy of Sciences - PNAS, 2012-10, Vol.109 (40), p.16063-16067 |
| issn | 0027-8424 1091-6490 |
| language | eng |
| recordid | cdi_pnas_primary_109_40_16063 |
| source | Jstor Complete Legacy; PubMed Central; Alma/SFX Local Collection; Free Full-Text Journals in Chemistry |
| subjects | College mathematics equations functional response models Generating function Integers Mathematical congruence Mathematical cusps Mathematical functions Mathematical sequences Mathematical theorems Physical Sciences Series convergence statistical analysis |
| title | Unimodal sequences and quantum and mock modular forms |
| url | https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-01-26T07%3A00%3A57IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-jstor_pnas_&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=Unimodal%20sequences%20and%20quantum%20and%20mock%20modular%20forms&rft.jtitle=Proceedings%20of%20the%20National%20Academy%20of%20Sciences%20-%20PNAS&rft.au=Bryson,%20Jennifer&rft.date=2012-10-02&rft.volume=109&rft.issue=40&rft.spage=16063&rft.epage=16067&rft.pages=16063-16067&rft.issn=0027-8424&rft.eissn=1091-6490&rft_id=info:doi/10.1073/pnas.1211964109&rft_dat=%3Cjstor_pnas_%3E41763202%3C/jstor_pnas_%3E%3Curl%3E%3C/url%3E&disable_directlink=true&sfx.directlink=off&sfx.report_link=0&rft_id=info:oai/&rft_pqid=1803145472&rft_id=info:pmid/&rft_jstor_id=41763202&rfr_iscdi=true |