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 
formula_1
Relation to other polynomials.
q-Hahn polynomials→ Quantum q-Krawtchouk polynomials：
formula_2
q-Hahn polynomials→ Hahn polynomials
make the substitutionformula_3,formula_4 into definition of q-Hahn polynomials, and find the limit q→1, we obtain
formula_5，which is exactly Hahn polynomials.