q-Hahn polynomials

In mathematics, the q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Template:Harvs 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} [\begin{matrix} q^{- n}, a b q^{n + 1}, q^{- x} \\ a q, q^{- N} \end{matrix}; q, q] .

$\lim_{a \to \infty} Q_{n} (q^{- x}; a; p, N | q) = K_{n}^{q t m} (q^{- x}; p, N; q)$

q-Hahn polynomials→ Hahn polynomials

make the substitution $α = q^{α}$ , $β = q^{β}$ 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)

，which is exactly Hahn polynomials.