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In mathematics, the ''q''-Krawtchouk polynomials are a family of basic hypergeometric
orthogonal polynomials In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product. The most widely used orthogonal polynomials are the cl ...
in the basic
Askey scheme In mathematics, the Askey scheme is a way of organizing orthogonal polynomials of hypergeometric or basic hypergeometric type into a hierarchy. For the classical orthogonal polynomials discussed in , the Askey scheme was first drawn by and by , ...
. give a detailed list of their properties. showed that the ''q''-Krawtchouk polynomials are spherical functions for 3 different
Chevalley group In mathematics, specifically in group theory, the phrase ''group of Lie type'' usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The phr ...
s 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 function In mathematics, basic hypergeometric series, or ''q''-hypergeometric series, are ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series ''x'n'' is called ...
s by :K_n(q^;p,N;q)=_3\phi_2\left begin q^,q^,-pq^n\\ q^,0\end ;q,q\right\quad n=0,1,2,...,N.


See also

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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 polynomi ...
*
dual q-Krawtchouk polynomials In mathematics, the dual ''q''-Krawtchouk polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties. Definition The polynomials are given in terms of basic hyperg ...
*
quantum q-Krawtchouk polynomials In mathematics, the quantum ''q''-Krawtchouk polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties. Definition The polynomials are given in terms of basic hy ...


Sources

* * * * * {{refend Orthogonal polynomials Q-analogs Special hypergeometric functions