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Discrete analogues of the Laguerre inequality. (English) Zbl 1064.26010

The (special Laguerre-Pólya) class \({\mathcal F}\) of real entire functions: \[ {\mathcal F} := \left\{f: \mathbb{R} \to \mathbb{R}; f(x) = cx^me^{\beta x} \prod_{k=1}^\omega {\left(1+\frac{x}{x_k}\right)~e^{-x/x_k}}, \quad \omega \leq \infty \right\} \] (\(c, \beta, x_k \in \mathbb{R}, m \in \mathbb{N} \cup \{0\}, \sum {x_k^{-2}} < \infty\)) is considered. For \(f \in \mathcal F\) we have the inequality \[ \sum_{j=-m}^m {(-1)^{m+j} \frac{f^{(m-j)}(x) f^{(m+j)}(x)}{(m-j)!(m+j)!}} \geq 0, \quad x \in \mathbb{R}, \tag{1} \] which is the classical Laguerre inequality when \(m = 1\).
The author proves the discrete analogue of (1): {If \(f \in \mathcal F\) and if the distance between the two consecutive zeros of \(f(x)\) is not smaller than \(\sqrt {\frac{4m+2}{m+2}}\) then (1) holds with the derivatives replaced by \(f(x-j)\) and \(f(x+j)\), respectively.} [For the case \(m=1\) cf. the author, J. Approximation Theory 111, No. 1, 31–49 (2001; Zbl 0993.33007).]
The inequality is useful in establishing sharp and explicit bounds for discrete orthogonal polynomials and of the distance between their extreme zeros. This is done, with the help of the package Mathematica, for Krawtchouk polynomials \(P_k\), which satisfy the difference equation \((n-x)P_k(x+1) = (n-2k)P_k(x) - xP_k(x-1), \quad x \in [0,n]\).

MSC:

26D05 Inequalities for trigonometric functions and polynomials
26E05 Real-analytic functions
39A12 Discrete version of topics in analysis
33C47 Other special orthogonal polynomials and functions

Citations:

Zbl 0993.33007

Software:

Mathematica
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References:

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