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 Roelof Koekoek, Hypergeometric Orthogonal Polynomials and its q-Analogues, p501,Springer,2010

: $K^_n (q^;p;N;q) = _3\phi_2\left( \begin q^,0,q^\\ pq,q^\end;q,q\right), \qquad n=0,1,2,\ldots, N.$

Relation to other polynomials

affine q-Krawtchouk polynomials → little q-Laguerre polynomials：

: $\lim_=K_n^\text(q^;p,N\mid q)=p_n(q^x;p,q)$.

References

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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, s2cid=119838893

Orthogonal polynomials
Q-analogs
Special hypergeometric functions