Generalized Krawtchouk polynomials: new properties. (English) Zbl 1055.33011

Sookoo, Norris

Arch. Math., Brno 36, No. 1, 9-16 (2000).

For positive integers \(n\) and \(q=p^m\) (\(p\) is a prime and \(m\) a positive integer) let \[ V(n,q)=\left \{(s_0,s_1,\dots ,s_{q-1}) \mid s_0,s_1,\dots ,s_{q-1}=0,1,\dots ,n \text{ with }\sum _{i=0}^{q-1}s_i=n \right \}. \] For \(\overline p=(p_0,p_1,\dots ,p_{q-1})\), \(\overline s=(s_0,s_1,\dots ,s_{q-1})\in V(n,q)\) the generalized
Krawtchouk polynomial \(K(\overline p,\overline s)\) is defined by \[ \sum _{r_{i,j}}\frac {\overline s!} {\overline r_0!\overline r_1!\dots \overline r_{q-1}!} \prod _{k=1}^{q-1}[\chi (\alpha ^{k-1})]^{R(k)}, \] where \(\alpha \) is a primitive element of the Galois field \(\mathrm{GF}(q)\) and \(\chi \) a character of \(\mathrm{GF}(q)\); the summation and other expressions are described in the article.
This polynomial was defined by F. J. MacWilliams, N. J. A. Sloane and J.-M. Goethals [Bell Syst. Tech. J. 51, 803–819 (1972; Zbl 0301.94010)] who gave some properties of it. This paper continues the investigation of these polynomials, especially orthogonality and recurrence relations are presented.

Reviewer: Ladislav Skula (Brno)

MSC:

33C45

Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)

Keywords:

orthogonality; recurrence relations; series

Citations:

Zbl 0301.94010