Affine q-Krawtchouk polynomials

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. Template:Harvs give a detailed list of their properties.

Definition

The polynomials are given in terms of basic hypergeometric functions and the Pochhammer symbol by

$K_{n}^{\text{aff}}(q^{-x};p;N;q)={_{2}\varphi _{1}}\left({\begin{matrix}q^{-n}&0&q^{-x}\\pq&q^{-N}\end{matrix}};q,q\right),\qquad n=0,1,2,\ldots ,N$

Relation to other polynomials

Affine q-Krawtchouk polynomials → Little q-Laguerre polynomials：

$\lim _{a\to 1}=K_{n}^{\text{aff}}(q^{x-N};p,N\mid q)=p_{n}(q^{x};p,q)$

References

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