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 (1989) harvtxt error: no target: CITEREFKoornwinder1989 (help) 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

${\displaystyle K_{n}(q^{-x};p,N;q)={}_{3}\phi _{2}\left[{\begin{matrix}q^{-n},q^{-x},-pq^{n}\\q^{-N},0\end{matrix}};q,q\right],\quad n=0,1,2,...,N.}$

See also[]

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

References[]

• Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 96 (2nd ed.), Cambridge University Press, doi:10.2277/0521833574, ISBN 978-0-521-83357-8, MR 2128719
• Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F. (2010), Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-642-05014-5, ISBN 978-3-642-05013-8, MR 2656096
• Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), http://dlmf.nist.gov/18 |contribution-url= missing title (help), in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248
• Sadjang, Patrick Njionou. Moments of Classical Orthogonal Polynomials (Ph.D.). Universität Kassel. Retrieved February 20, 2021.
• Stanton, Dennis (1981), "Three addition theorems for some q-Krawtchouk polynomials", Geometriae Dedicata, 10 (1): 403–425, doi:10.1007/BF01447435, ISSN 0046-5755, MR 0608153