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 by

$K_n^{\text{aff}}(q^{-x};p;N;q)={}_3{\varphi }_2(\begin{array}{c}{q}^{-n},0,q^{-x}\\ pq,q^{-N}\end{array};q,q),n=0,1,2,\dots ,N.$

Relation to other polynomials

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

$\underset{a\to 1}{\lim }=K_n^{\text{aff}}(q^{x-N};p,N\mid q)=p_n(q^x;p,q)$.

References

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