2017/06/30 by Chen, William Y. C., Jia, Dennis X. Q., Wang, Larry X. W. · 4 citations
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1706.10245
The Turán inequalities and the higher order Turán inequalities arise in the study of Maclaurin coefficients of an entire function in the Laguerre-Pólya class. A real sequence \an\ is said to satisfy the Turán inequalities if for n≥ 1, an2-an-1an+1≥ 0. It is said to satisfy the higher order Turán inequalities if for n≥ 1, 4(an2-an-1an+1)(an+12-anan+2)-(anan+1-an-1an+2)2≥ 0. A sequence satisfying the Turán inequalities is also called log-concave. 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 Chen that p(n) satisfies the higher order Turán inequalities for n≥ 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≥ 95, the Jensen polynomials g3,n-1(x)=p(n-1)+3p(n)x+3p(n+1)x2+p(n+2)x3 have only real zeros. We conjecture that for any positive integer m≥ 4 there exists an integer N(m) such that for n≥ N(m) , the polynomials ∑k=0m m\choose kp(n+k)xk have only real zeros. This conjecture was independently posed by Ono.