q-Krawtchouk polynomials
{{DISPLAYTITLE:q-Krawtchouk polynomials}}
In mathematics, the q-Krawtchouk polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme {{harvs|txt | last1=Koekoek | first1=Roelof | last2=Lesky | first2=Peter A. | last3=Swarttouw | first3=René F. | title=Hypergeometric orthogonal polynomials and their q-analogues | publisher=Springer-Verlag | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-642-05013-8 | doi=10.1007/978-3-642-05014-5 | mr=2656096 | year=2010|loc=14}}. give a detailed list of their properties.
{{harvtxt|Stanton|1981}} showed that the q-Krawtchouk polynomials are spherical functions for 3 different Chevalley groups over finite fields, and {{harvtxt|Koornwinder|Wong|Koekoek|Swarttouw|2010–2022}} 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
:
q^{-n},q^{-x},-pq^n\\
q^{-N},0\end{matrix}
;q,q\right],\quad n=0,1,2,...,N.
See also
Sources
{{refbegin}}
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| last1 = Koekoek | first1 = Roelof
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| last3 = Swarttouw | first3 = René F.
| year = 2010
| publisher = Springer-Verlag | location = Berlin, New York
| series = Springer Monographs in Mathematics
| doi = 10.1007/978-3-642-05014-5 | isbn = 978-3-642-05013-8 | mr = 2656096
}}
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| editor2-last = Lozier | editor2-first = Daniel M.
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- {{Citation| title = Three addition theorems for some q-Krawtchouk polynomials
| last = Stanton | first = Dennis | year = 1981
| journal = Geometriae Dedicata
| volume = 10 | issue = 1 | pages = 403–425
| doi = 10.1007/BF01447435 | issn = 0046-5755 | mr = 608153
| s2cid = 119838893 }}
{{refend}}