In mathematics, the dual ''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 class ...

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.

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 hy ...

s by :

K_n(\lambda(x);c,N, q)=_3\phi_2(q^,q^,cq^;q^,0, q;q),\quad n=0,1,2,...,N,

:where

\lambda(x)=q^+cq^.

References

* * *{{dlmf, id=18, title=Chapter 18: Orthogonal Polynomials, first=Tom H. , last=Koornwinder, first2=Roderick S. C., last2= Wong, first3=Roelof , last3=Koekoek, , first4=René F. , last4=Swarttouw Orthogonal polynomials Q-analogs Special hypergeometric functions