Krasikov, Ilia Discrete analogues of the Laguerre inequality. (English) Zbl 1064.26010 Anal. Appl., Singap. 1, No. 2, 189-197 (2003). 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 \((n-x)P_k(x+1) = (n-2k)P_k(x) - xP_k(x-1), \quad x \in [0,n]\). Reviewer: Bogdan Choczewski (Kraków) Cited in 1 ReviewCited in 4 Documents 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 Keywords:real entire functions; Laguerre-Pólya class; generalized Laguerre inequality; discrete orthogonal polynomials; Krawtchouk polynomials Citations:Zbl 0993.33007 Software:Mathematica PDFBibTeX XMLCite \textit{I. Krasikov}, Anal. Appl., Singap. 1, No. 2, 189--197 (2003; Zbl 1064.26010) Full Text: DOI arXiv References: [1] DOI: 10.1016/0021-9045(90)90072-X · Zbl 0693.33005 · doi:10.1016/0021-9045(90)90072-X [2] DOI: 10.2140/pjm.1989.136.241 · Zbl 0699.30007 · doi:10.2140/pjm.1989.136.241 [3] DOI: 10.1016/0196-8858(90)90013-O · Zbl 0707.11062 · doi:10.1016/0196-8858(90)90013-O [4] DOI: 10.1016/0022-247X(92)90025-9 · Zbl 0768.30005 · doi:10.1016/0022-247X(92)90025-9 [5] Foster W. H., Int. J. of Math. Algorithms 2 pp 121– [6] DOI: 10.1112/S1461157000000310 · Zbl 0970.33005 · doi:10.1112/S1461157000000310 [7] DOI: 10.1016/S0196-8858(02)00005-2 · Zbl 1018.33007 · doi:10.1016/S0196-8858(02)00005-2 [8] DOI: 10.1016/S0377-0427(98)00183-6 · Zbl 0948.41018 · doi:10.1016/S0377-0427(98)00183-6 [9] DOI: 10.1007/BF02422380 · JFM 43.0158.01 · doi:10.1007/BF02422380 [10] I. Krasikov, Codes and Association Schemes, AMS-DIMACS Volume series, eds. A. Barg and S. Litsyn (AMS, Providence, 2001) pp. 193–198. [11] DOI: 10.1006/jath.2001.3570 · Zbl 0993.33007 · doi:10.1006/jath.2001.3570 [12] Laguerre E., OEuvres 1 (1972) [13] DOI: 10.1006/jath.2000.3474 · Zbl 0963.33005 · doi:10.1006/jath.2000.3474 [14] Levin B. J., Tran. Math. Mono. 5, in: Distribution of Zeros of the Entire Functions (1964) [15] Love J. B., Amer. Math. Monthly 69 pp 668– [16] Nikiforov A. F., Classical Orthogonal Polynomials of a Discrete Variable (1985) · Zbl 0642.33020 [17] DOI: 10.1137/0502018 · Zbl 0212.40903 · doi:10.1137/0502018 [18] DOI: 10.2140/pjm.1973.44.675 · Zbl 0265.33012 · doi:10.2140/pjm.1973.44.675 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.