In mathematics, the quantum 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.
Definition The polynomials are given in terms of basic hypergeometric functions by
K n q t m ( q − x ; p , N ; q ) =
2 ϕ 1 [ q − n , q − x q − N ; q ; p q n + 1 ] n = 0 , 1 , 2 , . . . , N . {\displaystyle K_{n}^{qtm}(q^{-x};p,N;q)={}_{2}\phi _{1}\left[{\begin{matrix}q^{-n},q^{-x}\\q^{-N}\end{matrix}};q;pq^{n+1}\right]\qquad n=0,1,2,...,N.}
References 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, arXiv:math/9602214, doi:10.1007/978-3-642-05014-5, ISBN 978-3-642-05013-8, MR 2656096 Koekoek, Roelof; Swarttouw, René F. (1996), The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, arXiv:math/9602214, Bibcode:1996math......2214K Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "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.
