Q Hahn polynomials

Definition

The polynomials are given in terms of basic hypergeometric functions and the Pochhammer symbol by Q n ( x ; a , b , N ; q ) = 3 ϕ 2 [ q − n a b q n + 1 x a q q − N ; q , q ]

Relation to other polynomials

q-Hahn polynomials→ Quantum q-Krawtchouk polynomials：

lim a → ∞ 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.