Laguerre inequalities for discrete sequences

Abstract

The Turán inequality, the Laguerre inequality and their m-rd generalizations have been proved to be closely relative with the Laguerre-Pólya class and Riemann hypothesis. Since these two inequalities are equivalent to log-concavity of the discrete sequences, we consider whether their generalizations hold for discrete sequences. Recently, Chen, Jia and Wang proved that the partition function satisfies the Turán inequality of order 2 and thus the 3-rd Jensen polynomials associated with the partition function have only real zeros. Griffin, Ono, Rolen and Zagier proved the n-th Jensen polynomials associated with the Maclaurin coefficients of the function in the Laguerre-Pólya class and the partition function have only real zeros except finite terms. In this paper, we show the Laguerre inequality of order 2 is true for the partition function, the overpartition function, the Bernoulli numbers, the derangement numbers, the Motzkin numbers, the Fine numbers, the Franel numbers and the Domb numbers.