Orthogonality Relations for Multivariate Krawtchouk Polynomials

Mizukawa, Hiroshi

doi:10.3842/sigma.2011.017

Cited by 12 publications

(16 citation statements)

References 5 publications

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“…This section treats non-reversible chains using Biorthogonal expansions. Some examples of (Mizukawa, 2010(Mizukawa, , 2011 are treated. Let P = (P ij ) be a Markov transition matrix on [d] with stationary distribution {p i }.…”

Section: Non-reversible Chains and Biorthogonal Expansionsmentioning

confidence: 99%

“…Section 3.2 develops a non-reversible theory using bi-orthogonal expansions. This is applied to generalizations of an urn model of (Mizukawa, 2010(Mizukawa, , 2011. Section 3.3 offers further generalizations all of which are diagonalized by multivariate Krawtchouk polynomials.…”

Section: Markov Chains With Multivariate Krawtchouk Polynomial Eigenvmentioning

confidence: 99%

“…Example Mizukawa (2010Mizukawa ( , 2011 considered N balls distributed in d urns arranged around a circle and moved to another urn in one of three schemes:…”

Section: Non-reversible Chains and Biorthogonal Expansionsmentioning

confidence: 99%

“…This work captures chains previously studied by Hoare and Rahman (2008), Khare and Zhou (2009), Zhou and Lange (2009) and Mizukawa (2010), Mizukawa (2011). Multivariate Krawtchouk polynomials also have a universal quality, diagonalizing symmetrized products of general Markov chains.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

An introduction to multivariate Krawtchouk polynomials and their applications

Diaconis

Griffiths

2014

Journal of Statistical Planning and Inference

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Section: Non-reversible Chains and Biorthogonal Expansionsmentioning

confidence: 99%

Section: Markov Chains With Multivariate Krawtchouk Polynomial Eigenvmentioning

confidence: 99%

“…Example Mizukawa (2010Mizukawa ( , 2011 considered N balls distributed in d urns arranged around a circle and moved to another urn in one of three schemes:…”

Section: Non-reversible Chains and Biorthogonal Expansionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

An introduction to multivariate Krawtchouk polynomials and their applications

Diaconis

Griffiths

2014

Journal of Statistical Planning and Inference

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“…Recent representations and derivations of the orthogonality of these polynomials are in [3][4][5][6].…”

Section: Introductionmentioning

confidence: 99%

Multivariate Krawtchouk Polynomials and Composition Birth and Death Processes

Griffiths

2016

Symmetry

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This paper defines the multivariate Krawtchouk polynomials, orthogonal on the multinomial distribution, and summarizes their properties as a review. The multivariate Krawtchouk polynomials are symmetric functions of orthogonal sets of functions defined on each of N multinomial trials. The dual multivariate Krawtchouk polynomials, which also have a polynomial structure, are seen to occur naturally as spectral orthogonal polynomials in a Karlin and McGregor spectral representation of transition functions in a composition birth and death process. In this Markov composition process in continuous time, there are N independent and identically distributed birth and death processes each with support 0, 1, . . .. The state space in the composition process is the number of processes in the different states 0, 1, . . .. Dealing with the spectral representation requires new extensions of the multivariate Krawtchouk polynomials to orthogonal polynomials on a multinomial distribution with a countable infinity of states.

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Meixner polynomials in several variables satisfying bispectral difference equations

Илиев

2012

Advances in Applied Mathematics

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We construct a set M d whose points parametrize families of Meixner polynomials in d variables. There is a natural bispectral involution b on M d which corresponds to a symmetry between the variables and the degree indices of the polynomials. We define two sets of d commuting partial difference operators diagonalized by the polynomials. One of the sets consists of difference operators acting on the variables of the polynomials and the other one on their degree indices, thus proving their bispectrality. The two sets of partial difference operators are naturally connected via the involution b.

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Orthogonality Relations for Multivariate Krawtchouk Polynomials

Cited by 12 publications

References 5 publications

An introduction to multivariate Krawtchouk polynomials and their applications

An introduction to multivariate Krawtchouk polynomials and their applications

Multivariate Krawtchouk Polynomials and Composition Birth and Death Processes

Meixner polynomials in several variables satisfying bispectral difference equations

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