Continuous q-Jacobi polynomials

In mathematics, the continuous q-Jacobi polynomials PTemplate:Su(x|q), introduced by Template:Harvtxt, are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Template:Harvs give a detailed list of their properties.

Definition

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

$P_n^{(\alpha ,\beta )}(x;q)=\frac{(q^{n+1};q{)}_n}{(q;q{)}_n}{}_4{\varphi }_3[\begin{array}{c}{q}^{-n},q^{n+\alpha +\beta +1},q^{\frac{1}{2}\alpha +\frac{1}{4}{e}^{i\theta }},q^{\frac{1}{2}\alpha +\frac{1}{4}{e}^{-i\theta }}\\ q^{n+1},-q^{\frac{1}{2}(\alpha +\beta +1)},-q^{\frac{1}{2}(\alpha +\beta +2)}\end{array};q,q]x=\cos \theta .$

References

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