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

• Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed.), Cambridge University Press, ISBN 978-0-521-83357-8, MR 2128719
• Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F. (2010), Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-642-05014-5, ISBN 978-3-642-05013-8, MR 2656096
• Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Chapter 18: Orthogonal Polynomials", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.