A system of multivariable Krawtchouk polynomials and a probabilistic application

F. Alberto Grünbaum ; Mizan Rahman

Symmetry, integrability and geometry: methods and applications, Tome 7 (2011)

Résumé

The one variable Krawtchouk polynomials, a special case of the $_2F_1$ 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 $F_1$ 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.

Détail

Keywords: Gelfand–Aomoto hypergeometric functions, cumulative Bernoulli trial, poker dice.
Mots-clés : multivariable Krawtchouk polynomials

Bibliographie

[1] Andrews G., Askey R., Roy R., Special functions, Encyclopedia of Mathematics and its Applications, 71, Cambridge University Press, Cambridge, 1999 | MR | Zbl

[2] Aomoto K., Kita M., Hypergeometric functions, Springer, Berlin, 1994 (in Japanese)

[3] Geronimo J.S., Iliev P., “Bispectrality of multivariable Racah–Wilson poynomials”, Constr. Approx., 15 (2010), 417–457 ; arXiv: 0705.1469 | DOI | MR

[4] Gelfand I.M., “General theory of hypergeometric functions”, Soviet Math. Dokl., 33 (1986), 573–577 | MR

[5] Griffiths R.C., “Orthogonal polynomials on the multinomial distribution”, Austral. J. Statist., 13 (1971), 27–35 | DOI | MR | Zbl

[6] Grünbaum F.A., “Block tridiagonal matrices and a beefed-up version of the Ehrenfest urn model”, Modern Analysis and Applications, The Mark Krein Centenary Conference, v. 1, Oper. Theory Adv. Appl., 190, Operator Theory and Related Topics, Birkhäuser Verlag, Basel, 266–277 | DOI | MR

[7] Grünbaum F.A., Pacharoni I., Tirao J.A., Two stochastic models of a random walk in the $U(n)$-spherical duals of $U(n+1)$, arXiv: 1010.0720 | Zbl

[8] Hoare M.R., Rahman M., “Cumultive Bernoulli trials and Krawtchouk processes”, Stochastic Process. Appl., 16 (1983), 113–139 | DOI | MR

[9] Hoare M.R., Rahman M., “A probabilistic origin for a new class of bivariate polynomials”, SIGMA, 4 (2008), 089, 18 pp. ; arXiv: 0812.3879 | DOI | MR | Zbl

[10] Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental functions, v. I, II, III, McGraw-Hill Book Company, Inc., New York – Toronto – London, 1953, 1955

[11] Iliev P., Terwilliger P., The Rahman polynomials and the Lie algebra $sl_3(C)$, arXiv: 1006.5062

[12] Iliev P., A Lie theoretic interpretation of multivariate hypergeometric polynomials, arXiv: 1101.1683

[13] Mizukawa H., “Zonal spherical functions on the complex reflection groups and $(n+1,m+1)$-hypergeometric functions”, Adv. Math., 184 (2004), 1–17 | DOI | MR | Zbl

[14] Mizukawa H., “Orthogonality relations for multivariate Krawtchouck polynomials”, SIGMA, 7 (2011), 017, 5 pp. ; arXiv: 1009.1203 | DOI | MR | Zbl

[15] Mizukawa H., Tanaka H., “$(n+1,m+1)$-hypergeometric functions associated to character algebras”, Proc. Amer. Math. Soc., 132 (2004), 2613–2618 | DOI | MR | Zbl

[16] Tirao J.A., “The matrix-valued hypergeometric equation”, Proc. Natl. Acad. Sci. USA, 100:14 (2003), 8138–8141 | DOI | MR | Zbl