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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. give a detailed list of their properties.

Definition

The polynomials are given in terms of basic hypergeometric functions by

: K^{\text{aff}}_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), \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

References

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

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

orthogonal-polynomials q-analogs special-hypergeometric-functions
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