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 by

$Q_n(q^{-x};a,b,N;q)={}_3{\varphi }_2[\begin{array}{c}{q}^{-n},ab{q}^{n+1},q^{-x}\\ aq,q^{-N}\end{array};q,q].$

Relation to other polynomials

q-Hahn polynomials→ Quantum q-Krawtchouk polynomials：

$\underset{a\to \mathrm{\infty }}{\lim }{Q}_n(q^{-x};a;p,N|q)=K_n^{qtm}(q^{-x};p,N;q)$

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

${}_3{F}_2(-n,\alpha +\beta +n+1,-x,\alpha +1,-N,1)$，which is exactly Hahn polynomials.

References

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