In mathematics, the Askey–Wilson polynomials (or ''q''-Wilson polynomials) are a family of

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 ...

introduced by as

q-analog In mathematics, a ''q''-analog of a theorem, identity or expression is a generalization involving a new parameter ''q'' that returns the original theorem, identity or expression in the limit as . Typically, mathematicians are interested in ''q'' ...

s of the

Wilson polynomials In mathematics, Wilson polynomials are a family of orthogonal polynomials introduced by that generalize Jacobi polynomials, Hahn polynomials, and Charlier polynomials. They are defined in terms of the generalized hypergeometric function and the ...

. They include many of the other orthogonal polynomials in 1 variable as special or limiting cases, described in the

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 , ...

. Askey–Wilson polynomials are the special case of

Macdonald polynomials In mathematics, Macdonald polynomials ''P''λ(''x''; ''t'',''q'') are a family of orthogonal symmetric polynomials in several variables, introduced by Macdonald in 1987. He later introduced a non-symmetric generalization in 1995. Macdonald origin ...

(or

Koornwinder polynomials In mathematics, Macdonald-Koornwinder polynomials (also called Koornwinder polynomials) are a family of orthogonal polynomials in several variables, introduced by and I. G. Macdonald (1987, important special cases), that generalize the Askey–W ...

) for the non-reduced

affine root system In mathematics, an affine root system is a root system of affine-linear functions on a Euclidean space. They are used in the classification of affine Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g to ...

of type (), and their 4 parameters , , , correspond to the 4 orbits of roots of this root system. They are defined by :

p_n(x) =
p_n(x;a,b,c,d\mid q) :=
a^(ab,ac,ad;q)_n\;_\phi_3 \left begin 
q^&abcdq^&ae^&ae^ \\ 
ab&ac&ad \end 
; q,q \right

where is a

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 ...

, , and is the ''q''-Pochhammer symbol. Askey–Wilson functions are a generalization to non-integral values of .

Proof

This result can be proven since it is known that :

p_n(\cos) = p_n(\cos;a,b,c,d\mid q)

and using the definition of the ''q''-Pochhammer symbol :

p_n(\cos)=
a^\sum_^q^\left(abq^,acq^,adq^;q\right)_\times\frac\prod_^\left(1-2aq^j\cos+a^2q^\right)

which leads to the conclusion that it equals :

a^(ab,ac,ad;q)_n\;_\phi_3 \left begin 
q^&abcdq^&ae^&ae^ \\ 
ab&ac&ad \end
; q,q \right

References

* * * * Q-analogs Hypergeometric functions Orthogonal polynomials {{mathematics-stub

Proof

See also

References