In mathematics, the q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. txt, 14 give a detailed list of their properties.
Definition
The polynomials are given in terms of basic hypergeometric functions by
Q_n(q^{-x};a,b,N;q)={}_3\phi_2\left[\begin{matrix} q^{-n},abq^{n+1},q^{-x}\\ aq,q^{-N}\end{matrix} ;q,q\right].
Relation to other polynomials
q-Hahn polynomials→ Quantum q-Krawtchouk polynomials：
\lim_{a \to \infty}Q_{n}(q^{-x};a;p,N|q)=K_{n}^{qtm}(q^{-x};p,N;q)
q-Hahn polynomials→ Hahn polynomials
make the substitution\alpha=q^{\alpha},\beta=q^{\beta} into definition of q-Hahn polynomials, and find the limit q→1, we obtain
{}_3F_2(-n,\alpha+\beta+n+1,-x,\alpha+1,-N,1)，which is exactly Hahn polynomials.
References
- Gasper, George & Rahman, Mizan (2004), Basic hypergeometric series, Vol. 96, Encyclopedia of Mathematics and its Applications, 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
- 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