q-Hahn polynomials

See also: continuous q-Hahn polynomials, dual q-Hahn polynomials, and continuous dual q-Hahn polynomials

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 and the Pochhammer symbol by $Q_n(x;a,b,N;q)=\;_{3}\phi_2\left[\begin{matrix} q^-n & abq^n+1 & x \\ aq & q^-N \end{matrix} ; q,q \right]$

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.