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Q-Krawtchouk polynomials

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

showed that the q-Krawtchouk polynomials are spherical functions for 3 different Chevalley groups over finite fields, and showed that they are related to representations of the quantum group SU(2).

Definition

The polynomials are given in terms of basic hypergeometric functions by :K_n(q^{-x};p,N;q)={}_3\phi_2\left[\begin{matrix} q^{-n},q^{-x},-pq^n\ q^{-N},0\end{matrix} ;q,q\right],\quad n=0,1,2,...,N.

Sources

{{Citation| title = Basic hypergeometric series | edition = 2nd
{{Citation| title = Hypergeometric orthogonal polynomials and their q-analogues
{{Citation| contribution = Chapter 18 Orthogonal Polynomials | editor1-last = Olver | editor1-first = Frank W. J. | editor1-link = Frank W. J. Olver | editor2-last = Lozier | editor2-first = Daniel M. | editor3-last = Boisvert | editor3-first = Ronald F. | editor4-last = Clark | editor4-first = Charles W. | contribution-url = http://dlmf.nist.gov/18
{{cite thesis| type = Ph.D. thesis| title = Moments of Classical Orthogonal Polynomials
{{Citation| title = Three addition theorems for some q-Krawtchouk polynomials

Wikipedia Source

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orthogonal-polynomials q-analogs special-hypergeometric-functions

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