Exact formulae for the fractional partition functions
Abstract The partition function p(n) has been a testing ground for applications of analytic number theory to combinatorics. In particular, Hardy and Ramanujan invented the “circle method” to estimate the size of p(n), which was later perfected by Rademacher who obtained an exact formula. Recently,...
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| Format: | Article |
| Language: | English |
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Springer International Publishing
2021
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| Online Access: | https://hdl.handle.net/1721.1/131477 |
| _version_ | 1811069092628529152 |
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| author | Iskander, Jonas Jain, Vanshika Talvola, Victoria |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Iskander, Jonas Jain, Vanshika Talvola, Victoria |
| author_sort | Iskander, Jonas |
| collection | MIT |
| description | Abstract
The partition function p(n) has been a testing ground for applications of analytic number theory to combinatorics. In particular, Hardy and Ramanujan invented the “circle method” to estimate the size of p(n), which was later perfected by Rademacher who obtained an exact formula. Recently, Chan and Wang considered the fractional partition functions, defined for $$\alpha \in {\mathbb {Q}}$$α∈Q by $$\sum _{n = 0}^\infty p_{\alpha }(n)x^n := \prod _{k=1}^\infty (1-x^k)^{-\alpha }$$∑n=0∞pα(n)xn:=∏k=1∞(1-xk)-α. In this paper we use the Rademacher circle method to find an exact formula for $$p_\alpha (n)$$pα(n) and study its implications, including log-concavity and the higher-order generalizations (i.e., the Turán inequalities) that $$p_\alpha (n)$$pα(n) satisfies. |
| first_indexed | 2024-09-23T08:05:35Z |
| format | Article |
| id | mit-1721.1/131477 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T08:05:35Z |
| publishDate | 2021 |
| publisher | Springer International Publishing |
| record_format | dspace |
| spelling | mit-1721.1/1314772023-02-28T21:28:59Z Exact formulae for the fractional partition functions Iskander, Jonas Jain, Vanshika Talvola, Victoria Massachusetts Institute of Technology. Department of Mathematics Abstract The partition function p(n) has been a testing ground for applications of analytic number theory to combinatorics. In particular, Hardy and Ramanujan invented the “circle method” to estimate the size of p(n), which was later perfected by Rademacher who obtained an exact formula. Recently, Chan and Wang considered the fractional partition functions, defined for $$\alpha \in {\mathbb {Q}}$$α∈Q by $$\sum _{n = 0}^\infty p_{\alpha }(n)x^n := \prod _{k=1}^\infty (1-x^k)^{-\alpha }$$∑n=0∞pα(n)xn:=∏k=1∞(1-xk)-α. In this paper we use the Rademacher circle method to find an exact formula for $$p_\alpha (n)$$pα(n) and study its implications, including log-concavity and the higher-order generalizations (i.e., the Turán inequalities) that $$p_\alpha (n)$$pα(n) satisfies. 2021-09-20T17:17:14Z 2021-09-20T17:17:14Z 2020-04-21 2020-09-24T21:18:33Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/131477 Research in Number Theory. 2020 Apr 21;6(2):20 en https://doi.org/10.1007/s40993-020-00195-0 Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Springer Nature Switzerland AG application/pdf Springer International Publishing Springer International Publishing |
| spellingShingle | Iskander, Jonas Jain, Vanshika Talvola, Victoria Exact formulae for the fractional partition functions |
| title | Exact formulae for the fractional partition functions |
| title_full | Exact formulae for the fractional partition functions |
| title_fullStr | Exact formulae for the fractional partition functions |
| title_full_unstemmed | Exact formulae for the fractional partition functions |
| title_short | Exact formulae for the fractional partition functions |
| title_sort | exact formulae for the fractional partition functions |
| url | https://hdl.handle.net/1721.1/131477 |
| work_keys_str_mv | AT iskanderjonas exactformulaeforthefractionalpartitionfunctions AT jainvanshika exactformulaeforthefractionalpartitionfunctions AT talvolavictoria exactformulaeforthefractionalpartitionfunctions |