DISCRETE ANALOGUES OF THE LAGUERRE INEQUALITY

Abstract

It is shown that [Formula: see text], 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 that the distance between two consecutive zeros of f(x) is at least [Formula: see text]. These inequalities are surprisingly similar discrete analogues of higher degree generalizations of the Laguerre and Turán inequalities. Applied to the classical discrete orthogonal polynomials, they yield sharp, explicit bounds, uniform in all parameters involved, on the polynomials and their extreme zeros. This is illustrated for the case of Krawtchouk polynomials.