# .

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 Pochhammer symbol by [1]

$$p_n(x;a,b,c|q)=\frac{(ab,ac;q)_n}{a^n}*_3\Phi_2(q^-n,ae^{i\theta},ae^{-i\theta};ab,ac|q;q)$$

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

References

Mesuma Atakishiyeva,Natig Atakishieyev,A NON STANDARD GENERATING FUNCTION FOR CONTINUOS DUAL Q-HAHN POLYNOMIALS,REVISTA DE MATEMATICA 2011 18(1):111-120

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Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications 96 (2nd ed.), Cambridge University Press, doi:10.2277/0521833574, 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), http://dlmf.nist.gov/18 |contribution-url= missing title (help), in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248