q-Krawtchouk polynomials

In mathematics, the q-Krawtchouk polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14). give a detailed list of their properties.

Stanton (1981) showed that the q-Krawtchouk polynomials are spherical functions for 3 different Chevalley groups over finite fields, and Koornwinder et al. (2010–2022) showed that they are related to representations of the quantum group SU(2).

Definition

The polynomials are given in terms of basic hypergeometric functions by

$K_n(q^{-x};p,N;q)={}_3{\varphi }_2[\begin{array}{c}{q}^{-n},q^{-x},-p{q}^n\\ q^{-N},0\end{array};q,q],n=0,1,2,\mathrm{...},N.$

See also

• affine q-Krawtchouk polynomials
• dual q-Krawtchouk polynomials
• quantum q-Krawtchouk polynomials