Discrete Analogues of the Laguerre Inequality

Abstract

It is shown that

On zeros of discrete orthogonal polynomials

“…In the discrete case the only previously known results going beyond the first order bounds were obtained for the binary Krawtchouk and the Charlier polynomials [5,6]. Here we improve slightly these inequalities and establish similar results for the rest of the cases.…”

“…We also assume that A(x) and B(x) are continuous functions on L. It seems that the following easy to prove general result has been noticed only recently [5], although the corresponding statements for binary Krawtchouk and Hahn polynomials were known for a long time [1,12]. Theorem 1.…”

Bounds for Zeros of the Charlier Polynomials

“…Our main tools are the following inequalities established in [11], which are the discrete analogue of the generalization of the Laguerre inequality given by Jensen [7] and rediscovered by Patrick [15], [16].…”

“…First of all notice that by (10) one can express p(x + j) as a linear combination of p(x + 1) and p(x) with coefficients depending on x. Choosing either (11) or (14), depending on the mesh, and putting t = t(x) = p(x + 1)/p(x), one obtains a nonnegative expression…”

“…We shall use the approach developed in [4,10] and the difference analogues of Laguerre inequality found in [11]. Namely, we prove the following results.…”