An introduction to multivariate Krawtchouk polynomials and their applications

Diaconis, Persi; Griffiths, Robert

doi:10.1016/j.jspi.2014.02.004

Cited by 41 publications

(47 citation statements)

References 34 publications

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“…Note that for the large family of multivariable Krawtchouk polynomials introduced in [9], such Lie interpretation together with their spectral properties were obtained in [10]. For a recent introduction to these polynomials and numerous probabilistic applications, see [2] and the references therein.…”

Section: Introductionmentioning

confidence: 93%

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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: Introductionmentioning

confidence: 93%

“…Then the set {Y (1) ν : |ν| = n}∪{Y (2) ν : |ν| = n−1}, restricted to S d , is an orthogonal basis for H d+1 n (w κ ) (see, for example, [5,Section 4.2]). Comparing the above with Definition 9.1, we can add another parity at the last variable and establish the following result.…”

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confidence: 99%

Connection coefficients for classical orthogonal polynomials of several variables

Илиев

2017

Advances in Mathematics

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“…A recent introduction to them is Diaconis and Griffiths [2]. They play an important role in the spectral expansion of transition functions of composition Markov processes.…”

Section: Multivariate Krawtchouk Polynomialsmentioning

confidence: 99%

“…Griffiths [1] and Diaconis and Griffiths [2] construct multivariate Krawtchouk polynomials orthogonal on the multinomial distribution and study their properties. Recent representations and derivations of the orthogonality of these polynomials are in [3][4][5][6].…”

Section: Introductionmentioning

confidence: 99%

“…The approach of Diaconis and Griffiths [2] is probabilistic and directed to Markov chain applications; the approach of Iliev [5] is via Lie groups; and the physics approach of Genest et al [3] is as matrix elements of group representations on oscillator states. Xu [7] studies discrete multivariate orthogonal polynomials, which have a triangular construction of products of one-dimensional orthogonal polynomials.…”

Section: Introductionmentioning

confidence: 99%

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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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Spin Chains, Graphs and State Revival

Miki¹,

Tsujimoto

Vinet

2020

Orthogonal Polynomials

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Connections between the 1-excitation dynamics of spin lattices and quantum walks on graphs will be surveyed. Attention will be paid to perfect state transfer (PST) and fractional revival (FR) as well as to the role played by orthogonal polynomials in the study of these phenomena. Included is a discussion of the ordered Hamming scheme, its relation to multivariate Krawtchouk polynomials of the Tratnik type, the exploration of quantum walks on graphs of this association scheme and their projection to spin lattices with PST and FR. (2000). Primary 33C45; Secondary 81R30. Mathematics Subject Classification

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An introduction to multivariate Krawtchouk polynomials and their applications

Cited by 41 publications

References 34 publications

Connection coefficients for classical orthogonal polynomials of several variables

Connection coefficients for classical orthogonal polynomials of several variables

Multivariate Krawtchouk Polynomials and Composition Birth and Death Processes

Spin Chains, Graphs and State Revival

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