Discrete orthogonal polynomials and difference equations of several variables

Илиев, Пламен; Xu, Yuan

doi:10.1016/j.aim.2006.09.012

Cited by 35 publications

(48 citation statements)

References 11 publications

(10 reference statements)

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“…We note, however, that the polynomials in (8.3) differ by an unessential factor from the ones in [11]. With the current normalization, the Krawtchouk polynomials polynomials satisfy a simple duality relation.…”

Section: Krawtchouk Polynomials Of Several Variablesmentioning

confidence: 99%

“…Using the notation (2.5) for ρ, ν and x we can define a family K ν (·; ρ, N ) of the Krawtchouk polynomials as follows (see [11]). …”

Section: Krawtchouk Polynomials Of Several Variablesmentioning

confidence: 99%

“…The generating function was later identified as the Jacobi polynomials on the simplex and the basis was given explicitly; see [11,18] for further references.…”

Section: Hahn Polynomials Of Several Variablesmentioning

confidence: 99%

“…First of all, using the Jacobi polynomials on the simplex as generating functions of the Hahn polynomials [12,18], the connection coefficients for the Hahn polynomials of several variables and their permutations can be derived from those for the Jacobi polynomials on the simplex. Secondly, taking an appropriate limit ( [11,18]), we can derive explicit formulas for the connection coefficients of the Krawtchouk polynomials of several variables, which can be expressed by Krawtchouk polynomials with different parameters. Finally, it is known that the orthogonal polynomials on the unit ball and the unit sphere and those on the simplex are related, which can be used to derive connection coefficients for orthogonal bases on these domains.…”

Section: Introductionmentioning

confidence: 99%

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Connection coefficients for classical orthogonal polynomials of several variables

Илиев

2017

Advances in Mathematics

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Abstract. Connection coefficients between different orthonormal bases satisfy two discrete orthogonal relations themselves. For classical orthogonal polynomials whose weights are invariant under the action of the symmetric group, connection coefficients between a basis consisting of products of hypergeometric functions and another basis obtained from the first one by applying a permutation are studied. For the Jacobi polynomials on the simplex, it is shown that the connection coefficients can be expressed in terms of Tratnik's multivariable Racah polynomials and their weights. This gives, in particular, a new interpretation of the hidden duality between the variables and the degree indices of the Racah polynomials, which lies at the heart of their bispectral properties. These techniques also lead to explicit formulas for connection coefficients of Hahn and Krawtchouk polynomials of several variables, as well as for orthogonal polynomials on balls and spheres.

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Section: Krawtchouk Polynomials Of Several Variablesmentioning

confidence: 99%

“…Using the notation (2.5) for ρ, ν and x we can define a family K ν (·; ρ, N ) of the Krawtchouk polynomials as follows (see [11]). …”

Section: Krawtchouk Polynomials Of Several Variablesmentioning

confidence: 99%

“…The generating function was later identified as the Jacobi polynomials on the simplex and the basis was given explicitly; see [11,18] for further references.…”

Section: Hahn Polynomials Of Several Variablesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Connection coefficients for classical orthogonal polynomials of several variables

Илиев

2017

Advances in Mathematics

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“…He studied the second order partial difference equation of two variables to determine when it has orthogonal polynomials as solutions (Xu, 2005). Iliev and Xu (2007) found that second order difference equations have discrete orthogonal polynomials as their eigenfunctions. They provide a family of orthogonal basis explicitly, including Hahn polynomials of several variables.…”

Section: Introductionmentioning

confidence: 99%

Bivariate Hahn moments for image reconstruction

Wu¹,

Yan

2014

International Journal of Applied Mathematics and Computer Science

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This paper presents a new set of bivariate discrete orthogonal moments which are based on bivariate Hahn polynomials with non-separable basis. The polynomials are scaled to ensure numerical stability. Their computational aspects are discussed in detail. The principle of parameter selection is established by analyzing several plots of polynomials with different kinds of parameters. Appropriate parameters of binary images and a grayscale image are obtained through experimental results. The performance of the proposed moments in describing images is investigated through several image reconstruction experiments, including noisy and noise-free conditions. Comparisons with existing discrete orthogonal moments are also presented. The experimental results show that the proposed moments outperform slightly separable Hahn moments for higher orders.

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Gaudin Model for the Multinomial Distribution

Илиев

2023

Ann. Henri Poincaré

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Discrete orthogonal polynomials and difference equations of several variables

Abstract: The goal of this work is to characterize all second order difference operators of several variables that have discrete orthogonal polynomials as eigenfunctions. Under some mild assumptions, we give a complete solution of the problem.

Cited by 35 publications

References 11 publications

Connection coefficients for classical orthogonal polynomials of several variables

Connection coefficients for classical orthogonal polynomials of several variables

Bivariate Hahn moments for image reconstruction

Gaudin Model for the Multinomial Distribution

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