Dual q-Krawtchouk polynomials explained

In mathematics, the dual q-Krawtchouk polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.

Definition

The polynomials are given in terms of basic hypergeometric functions by

K _{n(λ(x);c,N|q)={}} _{3\phi} ^{- n} ^{2 (q} ,q ^{-x} ,cq ^{x-N} ;q ^{-N} , 0 |q;q), n =0 , 1 , 2 ,...,N,

where

λ (x) = q ^{-x} + cq ^{x-N} .