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| 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 | doi=10.1007/978-3-642-05014-5 | 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.
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| ==Definition==
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| The polynomials are given in terms of [[basic hypergeometric function]]s and the [[Pochhammer symbol]] by
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| :<math>\displaystyle </math> | |
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| ==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==
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| ==References==
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| *{{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}}
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| *{{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}}
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| *{{dlmf|id=18|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}}
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| *{{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}}
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| [[Category:Orthogonal polynomials]]
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| [[Category:Q-analogs]]
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| [[Category:Special hypergeometric functions]]
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