A System of Multivariable Krawtchouk Polynomials and a Probabilistic Application

• Дата: 2011
• Автори: Grünbaum, F.A., Rahman, M.
• Формат: Стаття
• Мова: English
• Опубліковано: Інститут математики НАН України 2011
• Назва видання: Symmetry, Integrability and Geometry: Methods and Applications
• Онлайн доступ: http://dspace.nbuv.gov.ua/handle/123456789/148084
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Цитувати: 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.

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.