Q-Krawtchouk polynomials
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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).
![{\displaystyle K_{n}(q^{-x};p,N;q)={}_{3}\phi _{2}\left[{\begin{matrix}q^{-n},q^{-x},-pq^{n}\\q^{-N},0\end{matrix}};q,q\right],\quad n=0,1,2,...,N.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb66d562dc6a06211468703e0706f773eab05a9f)