Quantum q-Krawtchouk polynomials

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

[math]\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. }[/math]

References

• 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, doi:10.1007/978-3-642-05014-5, ISBN 978-3-642-05013-8 
• Koekoek, Roelof; Swarttouw, René F. (1996), The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue 
• 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. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, http://dlmf.nist.gov/18