Affine q-Krawtchouk polynomials explained

In mathematics, the affine q-Krawtchouk polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by Carlitz and Hodges. give a detailed list of their properties.

Definition

The polynomials are given in terms of basic hypergeometric functions by ^[1]

	aff
K
	n

(q^-x;p;N;q)={}_3\phi_2\left(\begin{matrix}q^-n,0,q^-x\ pq,q^-N\end{matrix};q,q\right), n=0,1,2,\ldots,N.

Relation to other polynomials

affine q-Krawtchouk polynomials → little q-Laguerre polynomials：

\lim_a

	aff(q
=K
	n

^x-N;p,N\mid

	x;p,q)
q)=p
	n(q

Notes and References

Roelof Koekoek, Hypergeometric Orthogonal Polynomials and its q-Analogues, p. 501, Springer, 2010