Skip to main content Accessibility help

Determinantal inequalities for the partition function

  • Dennis X.Q. Jia (a1) and Larry X.W. Wang (a1)


Let p(n) denote the partition function. In this paper, we will prove that for $n\ges 222$ ,

$$\left| {\matrix{ {p(n)} & {p(n + 1)} & {p(n + 2)} \cr {p(n-1)} & {p(n)} & {p(n + 1)} \cr {p(n-2)} & {p(n-1)} & {p(n)} \cr } } \right| > 0.{\rm }$$
As a corollary, we deduce that p(n) satisfies the double Turán inequalities, that is, for $n\ges 222$ ,



Hide All
1Aissen, M., Edrei, A., Schoenberg, I. J. and Whitney, A. M.. On the generating functions of totally positive sequence. Proc. Nat. Acad. Sci. 37 (1951), 303307.
2Brenti, F.. Unimodal, log-concave, and Pólya frequency sequences in combinatorics. Mem. Amer. Math. Soc. 81 (1989).
3Chen, W. Y. C.. Recent developments on log-concavity and q-log-concavity of combinatorial polynomials. In FPSAC 2010 Conf. Talk Slides, (2010).
4Chen, W. Y. C., Jia, D. X. Q. and Wang, L. X. W.. Higher order Turán inequalities for the partition function. Trans. Amer. Math. Soc. (to appear),
5Chen, W. Y. C., Wang, L. X. W. and Xie, G. Y. B.. Finite differences of the logarithm of the partition function. Math. Comp. 85 (2016), 825847.
6Craven, T. and Csordas, G.. Jensen polynomials and the Turán and Laguerre inequalities. Pacific J. Math. 136 (1989), 241260.
7Craven, T. and Csordas, G.. Karlin's conjecture and a question of Pólya. Rocky Moutain J. Math. 35 (2005), 6182.
8Csordas, G. and Dimitrov, D. K.. Conjectures and theorems in the theory of entire functions. Numer. Alg. 25 (2000), 109122.
9DeSalvo, S. and Pak, I.. Log-concavity of the partition function. Ramanujan J. 38 (2015), 6173.
10Hou, Q. H. and Zhang, Z. R.. r-log-concavity of partition functions. Ramanujan J. (2018).
11Karlin, S.. Total positivity, vol. I (California: Standford University Press, 1968).
12Laguerre, E.. Oeuvres, vol. 1 (Paris: Gaauthier-Villars, 1989).
13Levin, B.Ja.. Distribution of zeros of entire functions, revised edn, Translations of Mathematical Monographs, vol. 5 (Providence, R.I.: American Mathematical Society, 1980).
14Pólya, G. and Schur, J.. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math. 144 (1914), 89113.
15Rahman, Q. I. and Schmeieer, G.. Analytic theroy of polynomials (Oxford: Oxford University Press, 2002).


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed