Higher order Turán inequalities for combinatorial sequences

Abstract

The Turán inequality and its higher order analog arise in the study of Maclaurin coefficients of an entire function in the Laguerre-Pólya class. It is well known that if a real entire function ψ(x) is in the LP class, the Maclaurin coefficients satisfy both the Turán inequality and the higher order Turán inequality. Chen, Jia and Wang proved that for n≥95, the higher order Turán inequality holds for the partition function p(n) and the 3-rd associated Jensen polynomials p(n)+3p(n+1)x+3p(n+2)x2+p(n+3)x3 have only real zeros. Recently, Griffin, Ono, Rolen and Zagier showed that Jensen polynomials for a large family of functions, including those associated to ξ(s) and the partition function, are hyperbolic for sufficiently large n. This result gave evidence for Riemann hypothesis. In this paper, we give a unified approach to investigate the higher order Turán inequality for the sequences {an/n!}n≥0, where an satisfy a three-term recurrence relation. In particular, we prove higher order Turán inequality for the sequences {an/n!}n≥0, where an are the Motzkin numbers, the Fine number, the Franel numbers of order 3 and the Domb numbers. As a consequence, for these combinatorial sequences, the 3-rd associated Jensen polynomialsann!+3an+1(n+1)!x+3an+2(n+2)!x2+an+3(n+3)!x3 have only real zeros. Furthermore, for these combinatorial sequences we conjecture that for any given integer m≥4, there exists an integer N(m) such that for n>N(m), the m-th associated Jensen polynomials∑i=0m(mi)an+i(n+i)!xi have only real zeros.