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

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.


33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)


Zbl 0301.94010
Full Text: EuDML