In mathematics, the ''q''-Hahn polynomials are a family of basic hypergeometric

orthogonal polynomials In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product. The most widely used orthogonal polynomials are the cl ...

in the basic

Askey scheme In mathematics, the Askey scheme is a way of organizing orthogonal polynomials of hypergeometric or basic hypergeometric type into a hierarchy. For the classical orthogonal polynomials discussed in , the Askey scheme was first drawn by and by , ...

. give a detailed list of their properties.

Definition

The polynomials are given in terms of

basic hypergeometric function In mathematics, basic hypergeometric series, or ''q''-hypergeometric series, are ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series ''x'n'' is called ...

s by :

Q_n(q^;a,b,N;q)=_3\phi_2\left begin 
q^,abq^,q^\\ 
aq,q^\end 
;q,q\right

Relation to other polynomials

q-Hahn polynomials→ Quantum q-Krawtchouk polynomials：

\lim_Q_(q^;a;p,N, q)=K_^(q^;p,N;q)

q-Hahn polynomials→

Hahn polynomials In mathematics, the Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials, introduced by Pafnuty Chebyshev in 1875 and rediscovered by Wolfgang Hahn . The Hahn class is a name for spe ...

make the substitution

\alpha=q^

\beta=q^

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

References

* * * *{{cite journal, last1=Costas-Santos, first1=R.S., last2=Sánchez-Lara, first2=J.F., title=Orthogonality of ''q''-polynomials for non-standard parameters, journal=Journal of Approximation Theory, date=September 2011, volume=163, issue=9, pages=1246–1268, doi=10.1016/j.jat.2011.04.005, arxiv=1002.4657, s2cid=115178147 Orthogonal polynomials Q-analogs Special hypergeometric functions