In mathematics, the 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 by
Q n ( q − x ; a , b , N ; q ) =
3 ϕ 2 [ q − n , a b q n + 1 , q − x a q , q − N ; q , q ] . {\displaystyle 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 → ∞ Q n ( q − x ; a ; p , N | q ) = K n q t m ( q − x ; p , N ; q ) {\displaystyle \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 α = q α {\displaystyle \alpha =q^{\alpha }} , β = q β {\displaystyle \beta =q^{\beta }} into definition of q-Hahn polynomials, and find the limit q→1, we obtain
3 F 2 ( − n , α + β + n + 1 , − x , α + 1 , − N , 1 ) {\displaystyle {}_{3}F_{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, 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.
