Q-Hahn polynomials

In mathematics, the q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. 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.