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Finite differences of the logarithm of the partition function


Authors: William Y. C. Chen, Larry X. W. Wang and Gary Y. B. Xie
Journal: Math. Comp. 85 (2016), 825-847
MSC (2010): Primary 05A20; Secondary 11B68
DOI: https://doi.org/10.1090/mcom/2999
Published electronically: July 22, 2015
MathSciNet review: 3434883
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Abstract: Let $ p(n)$ denote the partition function. DeSalvo and Pak proved that $ \frac {p(n-1)}{p(n)}\left (1+\frac {1}{n}\right )> \frac {p(n)}{p(n+1)}$ for $ n\geq 2$. Moreover, they conjectured that a sharper inequality $ \frac {p(n-1)}{p(n)}\left ( 1+\frac {\pi }{\sqrt {24}n^{3/2}}\right ) > \frac {p(n)}{p(n+1)}$ holds for $ n\geq 45$. In this paper, we prove the conjecture of Desalvo and Pak by giving an upper bound for $ -\Delta ^{2} \log p(n-1)$, where $ \Delta $ is the difference operator with respect to $ n$. We also show that for given $ r\geq 1$ and sufficiently large $ n$, $ (-1)^{r-1}\Delta ^{r} \log p(n)>0$. This is analogous to the positivity of finite differences of the partition function. It was conjectured by Good and proved by Gupta that for given $ r\geq 1$, $ \Delta ^{r} p(n)>0$ for sufficiently large $ n$.


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Additional Information

William Y. C. Chen
Affiliation: Center for Applied Mathematics, Tianjin University, Tianjin 300072, People’s Republic of China
Email: chenyc@tju.edu.cn

Larry X. W. Wang
Affiliation: Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 30071, People’s Republic of China
Email: wsw82@nankai.edu.cn

Gary Y. B. Xie
Affiliation: Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 30071, People’s Republic of China
Email: xieyibiao@mail.nankai.edu.cn

DOI: https://doi.org/10.1090/mcom/2999
Received by editor(s): July 1, 2014
Received by editor(s) in revised form: September 23, 2014
Published electronically: July 22, 2015
Additional Notes: The authors wish to thank the referee for helpful comments. This work was supported by the 973 Project, the PCSIRT Project of the Ministry of Education and the National Science Foundation of China.
Article copyright: © Copyright 2015 American Mathematical Society