In mathematics, the 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. ${\displaystyle R_{n}(q^{-x}+\gamma \delta q^{x+1},\gamma ,\delta ,N|q)={}_{3}\phi _{2}\left[{\begin{matrix}q^{-n},q^{-x},\gamma \delta q^{x+1}\\\gamma q,q^{-N}\end{matrix}};q,q\right],\quad n=0,1,2,...,N}$

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.
• Costas-Santos, R.S.; Sánchez-Lara, J.F. (September 2011). "Orthogonality of q-polynomials for non-standard parameters". Journal of Approximation Theory. 163 (9): 1246–1268. arXiv:1002.4657. doi:10.1016/j.jat.2011.04.005. S2CID 115178147.
• Sadjang, Patrick Njionou. Moments of Classical Orthogonal Polynomials (Ph.D.). Universität Kassel. CiteSeerX 10.1.1.643.3896.