Dicrete Analogues of the Laguerre Inequality

Abstract

It is shown that $\sum_{j=-m}^m (-1)^j \frac{f(x-j)(f(x+j)}{(m-j)! (m+j)!} \ge 0,$ $m=0,1,...,$ where $f(x)$ is either a real polynomial with only real zeros or an allied entire function of a special type, provided the distance between two consecutive zeros of $f(x)$ is at least $\sqrt{4-\frac{6}{m+2}}.$ These inequalities are a surprisingly similar discrete analogue of higher degree generalizations of the Laguerre and Turan inequalities. Being applied to the classical discrete orthogonal polynomials, they yield sharp, explicit bounds uniform in all parameters involved, on the polynomials and their extreme zeros. We will illustrate it for the case of Krawtchouk polynomials