q-Krawtchouk polynomials
From HandWiki
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 Wong) 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
- [math]\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. }[/math]
See also
Sources
- Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 96 (2nd ed.), Cambridge University Press, ISBN 978-0-521-83357-8
- 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
- Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010–2022), "Chapter 18 Orthogonal Polynomials", NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, http://dlmf.nist.gov/18
- 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
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