Continuous dual q-Hahn polynomials

In mathematics, the continuous dual q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. 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 q-Pochhammer symbol by

$p_n(x;a,b,c\mid q)=\frac{(ab,ac;q{)}_n}{a^n}{}_3{\varphi }_2(q^{-n},a{e}^{i\theta },a{e}^{-i\theta };ab,ac\mid q;q)$

In which $x=\cos (\theta )$

Gallery

File:Continuous dual qHahn function abs complex3D Maple PLOT.gif

File:Continuous dual qHahn function re complex3D Maple PLOT.gif

File:Continuous dual qHahn function Im complex3D Maple PLOT.gif

File:Continuous dual qHahn function RE density Maple PLOT.gif

File:Continuous dual qHahn function Im density Maple PLOT.gif

File:Continuous dual qHahn function ABS density Maple PLOT.gif

References

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