Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 7 (2011), 017, 5 pages      arXiv:1009.1203      http://dx.doi.org/10.3842/SIGMA.2011.017

Orthogonality Relations for Multivariate Krawtchouk Polynomials

Hiroshi Mizukawa
Department of Mathematics, National Defense Academy of Japan, Yokosuka 239-8686, Japan

Received September 08, 2010, in final form February 18, 2011; Published online February 22, 2011

Abstract
The orthogonality relations of multivariate Krawtchouk polynomials are discussed. In case of two variables, the necessary and sufficient conditions of orthogonality is given by Grünbaum and Rahman in [SIGMA 6 (2010), 090, 12 pages]. In this study, a simple proof of the necessary and sufficient condition of orthogonality is given for a general case.

Key words: multivariate orthogonal polynomial; hypergeometric function.

pdf (290 Kb)   tex (7 Kb)

References

  1. Griffiths R.C., Orthogonal polynomials on the multinomial distribution, Austral. J. Statist. 13 (1971), 27-35.
  2. Grünbaum F.A., Rahman M., On a family of 2-variable orthogonal Krawtchouk polynomials, SIGMA 6 (2010), 090, 12 pages, arXiv:1007.4327.
  3. Mizukawa H., Zonal spherical functions on the complex reflection groups and (m+1,n+1)-hypergeometric functions, Adv. Math. 184 (2004), 1-17.
  4. Mizukawa H., Tanaka H., (n+1,m+1)-hypergeometric functions associated to character algebras, Proc. Amer. Math. Soc. 132 (2004), 2613-2618.

Previous article   Next article   Contents of Volume 7 (2011)