Higher order Turán inequalities for the partition function

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 ≥ 0 \{a_{n}\}_{n\geq 0} of real numbers is said to satisfy the Turán inequalities or to be log-concave if for n ≥ 1 n\geq 1 , a n 2 − a n − 1 a n + 1 ≥ 0 a_n^2-a_{n-1}a_{n+1}\geq 0 . It is said to satisfy the higher order Turán inequalities if for n ≥ 1 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 ≥ 0 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 ) p(n) , DeSalvo and Pak showed that for n > 25 n>25 , the sequence { p ( n ) } n > 25 \{ p(n)\}_{n> 25} is log-concave, that is, p ( n ) 2 − p ( n − 1 ) p ( n + 1 ) > 0 p(n)^2-p(n-1)p(n+1)>0 for n > 25 n> 25 . It was conjectured by the first author that p ( n ) p(n) satisfies the higher order Turán inequalities for n ≥ 95 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 p(n+1)p(n-1)/p(n)^2 . Consequently, for n ≥ 95 n\geq 95 , the Jensen polynomials p ( n − 1 ) + 3 p ( n ) x + 3 p ( n + 1 ) x 2 + p ( n + 2 ) x 3 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 ≥ 4 m\geq 4 there exists an integer N ( m ) N(m) such that for n ≥ N ( m ) n\geq N(m) , the Jensen polynomial associated with the sequence ( p ( n ) , p ( n + 1 ) , … , p ( n + m ) ) (p(n),p(n+1),\ldots ,p(n+m)) has only real zeros. This conjecture was posed independently by Ono.