A System of Multivariable Krawtchouk Polynomials and a Probabilistic Application
The one variable Krawtchouk polynomials, a special case of the ₂F₁ function did appear in the spectral representation of the transition kernel for a Markov chain studied a long time ago by M. Hoare and M. Rahman. A multivariable extension of this Markov chain was considered in a later paper by these...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2011 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2011
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/148084 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | A System of Multivariable Krawtchouk Polynomials and a Probabilistic Application / F.A. Grünbaum, M. Rahman // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 16 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862658279640924160 |
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| author | Grünbaum, F.A. Rahman, M. |
| author_facet | Grünbaum, F.A. Rahman, M. |
| citation_txt | A System of Multivariable Krawtchouk Polynomials and a Probabilistic Application / F.A. Grünbaum, M. Rahman // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 16 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The one variable Krawtchouk polynomials, a special case of the ₂F₁ function did appear in the spectral representation of the transition kernel for a Markov chain studied a long time ago by M. Hoare and M. Rahman. A multivariable extension of this Markov chain was considered in a later paper by these authors where a certain two variable extension of the F₁ Appel function shows up in the spectral analysis of the corresponding transition kernel. Independently of any probabilistic consideration a certain multivariable version of the Gelfand-Aomoto hypergeometric function was considered in papers by H. Mizukawa and H. Tanaka. These authors and others such as P. Iliev and P. Tertwilliger treat the two-dimensional version of the Hoare-Rahman work from a Lie-theoretic point of view. P. Iliev then treats the general n-dimensional case. All of these authors proved several properties of these functions. Here we show that these functions play a crucial role in the spectral analysis of the transition kernel that comes from pushing the work of Hoare-Rahman to the multivariable case. The methods employed here to prove this as well as several properties of these functions are completely different to those used by the authors mentioned above.
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| first_indexed | 2025-12-02T07:34:30Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-148084 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-02T07:34:30Z |
| publishDate | 2011 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Grünbaum, F.A. Rahman, M. 2019-02-16T20:47:44Z 2019-02-16T20:47:44Z 2011 A System of Multivariable Krawtchouk Polynomials and a Probabilistic Application / F.A. Grünbaum, M. Rahman // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 16 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33C45; 22E46; 33C45; 60J35; 60J05 DOI: http://dx.doi.org/10.3842/SIGMA.2011.119 https://nasplib.isofts.kiev.ua/handle/123456789/148084 The one variable Krawtchouk polynomials, a special case of the ₂F₁ function did appear in the spectral representation of the transition kernel for a Markov chain studied a long time ago by M. Hoare and M. Rahman. A multivariable extension of this Markov chain was considered in a later paper by these authors where a certain two variable extension of the F₁ Appel function shows up in the spectral analysis of the corresponding transition kernel. Independently of any probabilistic consideration a certain multivariable version of the Gelfand-Aomoto hypergeometric function was considered in papers by H. Mizukawa and H. Tanaka. These authors and others such as P. Iliev and P. Tertwilliger treat the two-dimensional version of the Hoare-Rahman work from a Lie-theoretic point of view. P. Iliev then treats the general n-dimensional case. All of these authors proved several properties of these functions. Here we show that these functions play a crucial role in the spectral analysis of the transition kernel that comes from pushing the work of Hoare-Rahman to the multivariable case. The methods employed here to prove this as well as several properties of these functions are completely different to those used by the authors mentioned above. The research of the first author was supported in part by the Applied Math. Sciences subprogram of the Of fice of Energy Research, USDOE, under Contract DE-AC03-76SF00098. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A System of Multivariable Krawtchouk Polynomials and a Probabilistic Application Article published earlier |
| spellingShingle | A System of Multivariable Krawtchouk Polynomials and a Probabilistic Application Grünbaum, F.A. Rahman, M. |
| title | A System of Multivariable Krawtchouk Polynomials and a Probabilistic Application |
| title_full | A System of Multivariable Krawtchouk Polynomials and a Probabilistic Application |
| title_fullStr | A System of Multivariable Krawtchouk Polynomials and a Probabilistic Application |
| title_full_unstemmed | A System of Multivariable Krawtchouk Polynomials and a Probabilistic Application |
| title_short | A System of Multivariable Krawtchouk Polynomials and a Probabilistic Application |
| title_sort | system of multivariable krawtchouk polynomials and a probabilistic application |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/148084 |
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