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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Higher order Turán inequalities for the partition function


Authors: William Y. C. Chen, Dennis X. Q. Jia and Larry X. W. Wang
Journal: Trans. Amer. Math. Soc. 372 (2019), 2143-2165
MSC (2010): Primary 05A20, 11P82
DOI: https://doi.org/10.1090/tran/7707
Published electronically: December 26, 2018
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Abstract: The Turán inequalities and the higher order Turán inequalities arise in the study of the Maclaurin coefficients of real entire functions in the Laguerre-Pólya class. A sequence $ \{a_{n}\}_{n\geq 0}$ of real numbers is said to satisfy the Turán inequalities or to be log-concave if for $ n\geq 1$, $ a_n^2-a_{n-1}a_{n+1}\geq 0$. It is said to satisfy the higher order Turán inequalities if for $ n\geq 1$, $ 4(a_{n}^2-a_{n-1}a_{n+1})(a_{n+1}^2-a_{n}a_{n+2})-(a_{n}a_{n+1}-a_{n-1}a_{n+2})^2\geq 0$. For the partition function $ p(n)$, DeSalvo and Pak showed that for $ n>25$, the sequence $ \{ p(n)\}_{n> 25}$ is log-concave, that is, $ p(n)^2-p(n-1)p(n+1)>0$ for $ n> 25$. It was conjectured by the first author that $ p(n)$ satisfies the higher order Turán inequalities for $ n\geq 95$. In this paper, we prove this conjecture by using the Hardy-Ramanujan-Rademacher formula to derive an upper bound and a lower bound for $ p(n+1)p(n-1)/p(n)^2$. Consequently, for $ n\geq 95$, the Jensen polynomials $ p(n-1)+3p(n)x+3p(n+1)x^2+p(n+2)x^3$ have only distinct real zeros. We conjecture that for any positive integer $ m\geq 4$ there exists an integer $ N(m)$ such that for $ n\geq N(m)$, the Jensen polynomial associated with the sequence $ (p(n),p(n+1),\ldots ,p(n+m))$ has only real zeros. This conjecture was posed independently by Ono.


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Additional Information

William Y. C. Chen
Affiliation: Center for Combinatorics, LPMC, Nankai University, Tianjin 300071, People’s Republic of China; and Center for Applied Mathematics, Tianjin University, Tianjin 300072, People’s Republic of China
Email: chen@nankai.edu.cn

Dennis X. Q. Jia
Affiliation: Center for Combinatorics, LPMC, Nankai University, Tianjin 300071, People’s Republic of China
Email: dennisjxq@mail.nankai.edu.cn

Larry X. W. Wang
Affiliation: Center for Combinatorics, LPMC, Nankai University, Tianjin 300071, People’s Republic of China
Email: wsw82@nankai.edu.cn

DOI: https://doi.org/10.1090/tran/7707
Keywords: Partition function, log-concavity, higher order Tur\'an inequalities, Hardy--Ramanujan--Rademacher formula, Jensen polynomials
Received by editor(s): August 5, 2017
Received by editor(s) in revised form: September 19, 2018
Published electronically: December 26, 2018
Additional Notes: This work was supported by the National Science Foundation of China.
Article copyright: © Copyright 2018 American Mathematical Society