q-Racah polynomials

In mathematics, the q-Racah polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by Askey & Wilson (1979). Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.

Definition

The polynomials are given in terms of basic hypergeometric functions and the Pochhammer symbol by

$p_n(q^{-x}+q^{x+1}cd;a,b,c,d;q)={}_4{\varphi }_3[\begin{array}{cccc}{q}^{-n}& ab{q}^{n+1}& q^{-x}& q^{x+1}cd\\ aq& bdq& cq\end{array};q;q]$

They are sometimes given with changes of variables as

$W_n(x;a,b,c,N;q)={}_4{\varphi }_3[\begin{array}{cccc}{q}^{-n}& ab{q}^{n+1}& q^{-x}& c{q}^{x-n}\\ aq& bcq& q^{-N}\end{array};q;q]$

Relation to other polynomials

q-Racah polynomials→Racah polynomials

References

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