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Keywords = classical Krawtchouk orthogonal polynomials

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27 pages, 511 KiB  
Article
On Perfectness of Systems of Weights Satisfying Pearson’s Equation with Nonstandard Parameters
by Alexander Aptekarev, Alexander Dyachenko and Vladimir Lysov
Axioms 2023, 12(1), 89; https://doi.org/10.3390/axioms12010089 - 15 Jan 2023
Cited by 2 | Viewed by 1366
Abstract
Measures generating classical orthogonal polynomials are determined by Pearson’s equation, whose parameters usually provide the positivity of the measures. The case of general complex parameters (nonstandard) is also of interest; the non-Hermitian orthogonality with respect to (now complex-valued) measures is considered on curves [...] Read more.
Measures generating classical orthogonal polynomials are determined by Pearson’s equation, whose parameters usually provide the positivity of the measures. The case of general complex parameters (nonstandard) is also of interest; the non-Hermitian orthogonality with respect to (now complex-valued) measures is considered on curves in C. Some applications lead to multiple orthogonality with respect to a number of such measures. For a system of r orthogonality measures, the perfectness is an important property: in particular, it implies the uniqueness for the whole family of corresponding multiple orthogonal polynomials and the (r+2)-term recurrence relations. In this paper, we introduce a unified approach which allows to prove the perfectness of the systems of complex measures satisfying Pearson’s equation with nonstandard parameters. We also study the polynomials satisfying multiple orthogonality relations with respect to a system of discrete measures. The well-studied families of multiple Charlier, Krawtchouk, Meixner and Hahn polynomials correspond to the systems of measures defined by the difference Pearson’s equation with standard real parameters. Using the same approach, we verify the perfectness of such systems for general parameters. For some values of the parameters, discrete measures should be replaced with the continuous measures with non-real supports. Full article
(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications)
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20 pages, 947 KiB  
Article
Watermarking Applications of Krawtchouk–Sobolev Type Orthogonal Moments
by Edmundo J. Huertas, Alberto Lastra and Anier Soria-Lorente
Electronics 2022, 11(3), 500; https://doi.org/10.3390/electronics11030500 - 8 Feb 2022
Cited by 2 | Viewed by 1648
Abstract
In this contribution, we consider the sequence {Hn(x;q)}n0 of monic polynomials orthogonal with respect to a Sobolev-type inner product involving forward difference operators For the first time in the literature, we apply [...] Read more.
In this contribution, we consider the sequence {Hn(x;q)}n0 of monic polynomials orthogonal with respect to a Sobolev-type inner product involving forward difference operators For the first time in the literature, we apply the non-standard properties of {Hn(x;q)}n0 in a watermarking problem. Several differences are found in this watermarking application for the non-standard cases (when j>0) with respect to the standard classical Krawtchouk case λ=μ=0. Full article
(This article belongs to the Special Issue Recent Developments and Applications of Image Watermarking)
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9 pages, 326 KiB  
Article
Stable Calculation of Krawtchouk Functions from Triplet Relations
by Albertus C. den Brinker
Mathematics 2021, 9(16), 1972; https://doi.org/10.3390/math9161972 - 18 Aug 2021
Cited by 4 | Viewed by 1606
Abstract
Deployment of the recurrence relation or difference equation to generate discrete classical orthogonal polynomials is vulnerable to error propagation. This issue is addressed for the case of Krawtchouk functions, i.e., the orthonormal basis derived from the Krawtchouk polynomials. An algorithm is proposed for [...] Read more.
Deployment of the recurrence relation or difference equation to generate discrete classical orthogonal polynomials is vulnerable to error propagation. This issue is addressed for the case of Krawtchouk functions, i.e., the orthonormal basis derived from the Krawtchouk polynomials. An algorithm is proposed for stable determination of these functions. This is achieved by defining proper initial points for the start of the recursions, balancing the order of the direction in which recursions are executed and adaptively restricting the range over which equations are applied. The adaptation is controlled by a user-specified deviation from unit norm. The theoretical background is given, the algorithmic concept is explained and the effect of controlled accuracy is demonstrated by examples. Full article
(This article belongs to the Section Mathematics and Computer Science)
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620 KiB  
Article
On Sister Celine’s Polynomials of Several Variables
by HSP Shrivastava
Math. Comput. Appl. 2004, 9(2), 309-320; https://doi.org/10.3390/mca9020309 - 1 Aug 2004
Cited by 1 | Viewed by 1215
Abstract
The aim of the present paper is to define Sister Celine's polynomials of two and more variables. We reduce the two variables Sister Celine's polynomials into many classical orthogonal polynomials and their product also such as – Jacobi, Gegenbauer, Legendre, Laguerre, Bessel and [...] Read more.
The aim of the present paper is to define Sister Celine's polynomials of two and more variables. We reduce the two variables Sister Celine's polynomials into many classical orthogonal polynomials and their product also such as – Jacobi, Gegenbauer, Legendre, Laguerre, Bessel and some discrete polynomials Bateman, Pasternak, Hahn, Krawtchouk, Meixner, Poisson-Charlier & others. Many integral representations and generating function relations are also established. Full article
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