In mathematics, the dual 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.
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Definition
The polynomials are given in terms of basic hypergeometric functions by
K n ( λ ( x ) ; c , N | q ) = 3 ϕ 2 ( q − n , q − x , c q x − N ; q − N , 0 | q ; q ) , n = 0 , 1 , 2 , . . . , N , {\displaystyle K_{n}(\lambda (x);c,N|q)={}_{3}\phi _{2}(q^{-n},q^{-x},cq^{x-N};q^{-N},0|q;q),\quad n=0,1,2,...,N,} where λ ( x ) = q − x + c q x − N . {\displaystyle \lambda (x)=q^{-x}+cq^{x-N}.}- 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), "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.