View source for 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. {{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.

==Definition==

The  polynomials are given in terms of [[basic hypergeometric function]]s and the [[Pochhammer symbol]] by <ref>Roelof Koekoek, Hypergeometric Orthogonal Polynomials and its q-Analogues, p501,Springer,2010</ref>

: <math> K^{\text{aff}}_n (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</math>

==Orthogonality==
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==Recurrence and difference relations==
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==Rodrigues formula==
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==Generating function==
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==Relation to other polynomials==

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

: <math>\lim_{a \to 1}=K_n^\text{aff}(q^{x-N};p,N\mid q)=p_n(q^x;p,q)</math>

==References==
{{Reflist}}

*{{Citation | last1=Gasper | first1=George | last2=Rahman | first2=Mizan | title=Basic hypergeometric series | publisher=[[Cambridge University Press]] | edition=2nd | series=Encyclopedia of Mathematics and its Applications | isbn=978-0-521-83357-8 | doi=10.2277/0521833574 | mr=2128719 | year=2004 | volume=96}}
*{{Citation | 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}}
*{{dlmf|id=18|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}}
*{{Citation | last1=Stanton | first1=Dennis | title=Three addition theorems for some q-Krawtchouk polynomials | doi=10.1007/BF01447435 | mr=608153 | year=1981 | journal=Geometriae Dedicata | issn=0046-5755 | volume=10 | issue=1 | pages=403–425}}

[[Category:Orthogonal polynomials]]
[[Category:Q-analogs]]
[[Category:Special hypergeometric functions]]