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

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

See also

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

Sources

• Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed.), Cambridge University Press, 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. (2010–2022), "Chapter 18 Orthogonal Polynomials", 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 (n.d.). Moments of Classical Orthogonal Polynomials (Ph.D. thesis). Universität Kassel. CiteSeerX 10.1.1.643.3896.
• 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, S2CID 119838893