Dual q-Krawtchouk polynomials

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)^[1] give a detailed list of their properties.

Definition

The polynomials are given in terms of basic hypergeometric functions by

${\displaystyle K_{n}(\lambda (x);c,N|q)={}_{3}\phi _{2}(q^{-n},q^{-x},cq^{x-N};q^{-N},0|q;q),\quad n=0,1,2,...,N,}$

where ${\displaystyle \lambda (x)=q^{-x}+cq^{x-N}.}$