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§18.28 Askey–Wilson Class

  1. §18.28(i) Introduction
  2. §18.28(ii) Askey–Wilson Polynomials
  3. §18.28(iii) Al-Salam–Chihara Polynomials
  4. §18.28(iv) q1-Al-Salam–Chihara Polynomials
  5. §18.28(v) Continuous q-Ultraspherical Polynomials
  6. §18.28(vi) Continuous q-Hermite Polynomials
  7. §18.28(vii) Continuous q1-Hermite Polynomials
  8. §18.28(viii) q-Racah Polynomials
  9. §18.28(ix) Continuous q-Jacobi Polynomials
  10. §18.28(x) Limit Relations
  11. §18.28(xi) Limits for q1

§18.28(i) Introduction

The Askey–Wilson class OP’s comprise the four-parameter families of Askey–Wilson polynomials and of q-Racah polynomials, and cases of these families obtained by specialization of parameters. The Askey–Wilson polynomials form a system of OP’s {pn(x)}, n=0,1,2,, that are orthogonal with respect to a weight function on a bounded interval, possibly supplemented with discrete weights on a finite set. The q-Racah polynomials form a system of OP’s {pn(x)}, n=0,1,2,,N, that are orthogonal with respect to a weight function on a sequence {qy+cqy+1}, y=0,1,,N, with c a constant. Both the Askey–Wilson polynomials and the q-Racah polynomials can best be described as functions of z (resp. y) such that Pn(z)=pn(12(z+z1)) in the Askey–Wilson case, and Pn(y)=pn(qy+cqy+1) in the q-Racah case, and both are eigenfunctions of a second order q-difference operator similar to (18.27.1).

In the remainder of this section the Askey–Wilson class OP’s are defined by their q-hypergeometric representations, followed by their orthogonal properties. For further properties see Koekoek et al. (2010, Chapter 14). See also Gasper and Rahman (2004, pp. 180–199), Ismail (2009, Chapter 15), and Koornwinder (2012). For the notation of q-hypergeometric functions see §§17.2 and 17.4(i).

§18.28(ii) Askey–Wilson Polynomials

18.28.1 pn(x)=pn(x;a,b,c,d|q)=an=0nq(abq,acq,adq;q)n×(qn,abcdqn1;q)(q;q)×j=01(12aqjx+a2q2j),
18.28.1_5 Rn(z)=Rn(z;a,b,c,d|q)=pn(12(z+z1);a,b,c,d|q)an(ab,ac,ad;q)n=ϕ34(qn,abcdqn1,az,az1ab,ac,ad;q,q).

The polynomials pn(x;a,b,c,d|q) are symmetric in the parameters a,b,c,d.


Assume a,b,c,d are all real, or two of them are real and two form a conjugate pair, or none of them are real but they form two conjugate pairs.

18.28.2 11pn(x)pm(x)w(x)dx=hnδn,m,
|a|,|b|,|c|,|d|1, ab,ac,ad,bc,bd,cd1,


18.28.3 2πsinθw(cosθ)=|(e2iθ;q)(aeiθ,beiθ,ceiθ,deiθ;q)|2,
18.28.4 h0=(abcd;q)(q,ab,ac,ad,bc,bd,cd;q),
18.28.5 hn=h0(1abcdqn1)(q,ab,ac,ad,bc,bd,cd;q)n(1abcdq2n1)(abcd;q)n,

More generally,

18.28.6 11pn(x)pm(x)w(x)dx+pn(x)pm(x)ω=hnδn,m,

with w(x) and hn as above. Also, x are the points 12(αq+α1q) with α any of the a,b,c,d whose absolute value exceeds 1, and the sum is over the =0,1,2, with |αq|>1. See Koekoek et al. (2010, Eq. (14.1.3)) for the value of ω when α=a.

q-Difference Equation

18.28.6_1 (LRn)(z)=(qn+abcdqn1)Rn(z),

where the operator L is defined by

18.28.6_2 (Lf)(z)=(1az)(1bz)(1cz)(1dz)(1z2)(1qz2)(f(qz)f(z))+(1az1)(1bz1)(1cz1)(1dz1)(1z2)(1qz2)×(f(q1z)f(z))+(1+q1abcd)f(z).

Recurrence Relation

18.28.6_3 (z+z1)Rn(z)=an(Rn+1(z)Rn(z))+cn(Rn1(z)Rn(z))+(a+a1)Rn(z),

where c0=0 and

18.28.6_4 an =(1abqn)(1acqn)(1adqn)(1abcdqn1)a(1abcdq2n1)(1abcdq2n),
cn =a(1qn)(1bcqn1)(1bdqn1)(1cdqn1)(1abcdq2n2)(1abcdq2n1).


Define dual parameters a~,b~,c~,d~ in terms of a,b,c,d by


assuming a,b,c,d0. Then

18.28.6_5 Rn(a1qm;a,b,c,d|q)=Rm(a~1qn;a~,b~,c~,d~|q),

§18.28(iii) Al-Salam–Chihara Polynomials

18.28.7 Qn(cosθ;a,b|q)=pn(cosθ;a,b,0,0|q)=an=0nq(abq;q)n(qn;q)(q;q)j=01(12aqjcosθ+a2q2j)=(ab;q)nanϕ23(qn,aeiθ,aeiθab,0;q,q)=(beiθ;q)neinθϕ12(qn,aeiθb1q1neiθ;q,b1qeiθ).
18.28.8 12π0πQn(cosθ;a,b|q)Qm(cosθ;a,b|q)|(e2iθ;q)(aeiθ,beiθ;q)|2dθ=δn,m(qn+1,abqn;q),
a,b or a=b¯; ab1; |a|,|b|1.

More generally, if |ab|1 instead of |a|,|b|1, discrete terms need to be added to the right-hand side of (18.28.8); see Koekoek et al. (2010, Eq. (14.8.3)).

§18.28(iv) q1-Al-Salam–Chihara Polynomials

18.28.9 Qn(12(aqy+a1qy);a,b|q1)=(1)nbnq12n(n1)((ab)1;q)nϕ13(qn,qy,a2qy(ab)1;q,qnab1).
18.28.10 y=0(1q2ya2)(a2,(ab)1;q)y(1a2)(q,bqa1;q)y(ba1)y×qy2Qn(12(aqy+a1qy);a,b|q1)×Qm(12(aqy+a1qy);a,b|q1)=(qa2;q)(ba1q;q)(q,(ab)1;q)n(ab)nqn2δn,m.

Eq. (18.28.10) is valid when either

18.28.11 0<q<1,a,b,ab>1,a1b<q1,


18.28.12 0<q<1,a/i,b/i,(a)(b)>0,a1b<q1.

If, in addition to (18.28.11) or (18.28.12), we have a1bq, then the measure in (18.28.10) is the unique orthogonality measure. Also, if q<a1b<q1, then (18.28.10) holds with a,b interchanged. For further nondegenerate cases see Chihara and Ismail (1993) and Christiansen and Ismail (2006).

§18.28(v) Continuous q-Ultraspherical Polynomials

18.28.13 Cn(cosθ;β|q) ==0n(β;q)(β;q)n(q;q)(q;q)nei(n2)θ=(β;q)n(q;q)neinθϕ12(qn,ββ1q1n;q,β1qe2iθ).
18.28.14 Cn(cosθ;β|q) =(β2;q)n(q;q)nβ12nϕ34(qn,β2qn,β12eiθ,β12eiθβq12,β,βq12;q,q).
18.28.15 12π0πCn(cosθ;β|q)Cm(cosθ;β|q)|(e2iθ;q)(βe2iθ;q)|2dθ=(β,βq;q)(β2,q;q)(1β)(β2;q)n(1βqn)(q;q)nδn,m,

These polynomials are also called Rogers polynomials.

§18.28(vi) Continuous q-Hermite Polynomials

§18.28(vii) Continuous q1-Hermite Polynomials

18.28.18 hn(sinht|q)==0nq12(+1)(qn;q)(q;q)e(n2)t=entϕ11(qn0;q,qe2t)=inHn(isinht|q1).

For continuous q1-Hermite polynomials the orthogonality measure is not unique. See Askey (1989) and Ismail and Masson (1994) for examples.

§18.28(viii) q-Racah Polynomials

With x=qy+γδqy+1,

18.28.19 Rn(x)=Rn(x;α,β,γ,δ|q)==0nq(qn,αβqn+1;q)(αq,βδq,γq,q;q)j=01(1qjx+γδq2j+1)=ϕ34(qn,αβqn+1,qy,γδqy+1αq,βδq,γq;q,q),
αq, βδq, or γq=qN; n=0,1,,N.
18.28.20 y=0NRn(qy+γδqy+1)Rm(qy+γδqy+1)ωy=hnδn,m,


18.28.21 ωy=(αq,βδq,γq,γδq;q)y(q,γδαq,γβq,δq;q)y1γδq2y+1(αβq)y,
18.28.22 hn=(αβ)n+1q(n+1)2αβq2n+11(q;q)n(αq,βδq,γq;q)n(γαβqn,δαqn,1βqn,γδq;q)(1αβqn,γδαq,γβq,δq;q).


18.28.23 Rn(qy+γδqy+1;α,β,γ,δ|q)=Ry(qn+αβqn+1;γ,δ,α,β|q),
αq, βδq, or γq=qN; n,y=0,1,,N.

Leonard (1982) classified all (finite or infinite) discrete systems of OP’s pn(x) on a set {x(m)} for which there is a system of discrete OP’s qm(y) on a set {y(n)} such that pn(x(m))=qm(y(n)). These systems are the q-Racah polynomials and its limit cases.

§18.28(ix) Continuous q-Jacobi Polynomials

Let x=12(z+z1) and let Rn(z) be given by (18.28.1_5). The continuous q-Jacobi polynomial Pn(α,β)(x|q) is defined by

18.28.24 (q;q)n(qα+1;q)nPn(α,β)(x|q)=Rn(z;q12α+14,q12α+34,q12β+14,q12β+34|q)=ϕ34(qn,qn+α+β+1,q12α+14z,q12α+14z1qα+1,q12(α+β+1),q12(α+β+2);q,q).

Specialization to continuous q-ultraspherical:

18.28.25 Pn(λ12,λ12)(x|q)=q12nλ(qλ+1/2;q)n(q2λ;q)nCn(x;qλ|q).

§18.28(x) Limit Relations

Let Rn(z) be as defined in (18.28.1_5) and put rn(x;a,b,c,d|q)=Rn(z;a,b,c,d|q), x=12(z+z1).

From Askey–Wilson to Big q-Jacobi

18.28.26 limλ0rn(x/(2λ);λ,qaλ1,qcλ1,bc1λ|q)=Pn(x;a,b,c;q).

From Askey–Wilson to Little q-Jacobi

18.28.27 limλ0rn(bqx/(2λ);λ,qbλ1,q,a|q)=(b)nqn(n+1)/2(qa;q)n(qb;q)npn(x;a,b;q).
18.28.28 limμ0,λ/μ0rn(x/(2λμ);λ/μ,qaμ/λ,1/(λμ),qbλμ|q)=pn(x;a,b;q).

From Askey–Wilson to Wilson

18.28.29 limq1pn(112x(1q)2;qa,qb,qc,qd|q)(1q)3n=Wn(x;a,b,c,d).

From Continuous q-Jacobi to Jacobi

18.28.30 limq1Pn(α,β)(x|q)=Pn(α,β)(x).

From Continuous q-Ultraspherical to Ultraspherical

18.28.31 limq1Cn(x;qλ|q)=Cn(λ)(x).

From Continuous q-Ultraspherical to Continuous q-Hermite

18.28.32 limβ0Cn(x;β|q)=Hn(x|q)(q;q)n.

From Continuous q-Hermite to Hermite

18.28.33 limq1Hn(x(1q)/2|q)((1q)/2)n/2=Hn(x).

From q-Racah to Racah

18.28.34 limq1Rn(qy+qy+γ+δ+1;qα,qβ,qγ,qδ|q)=Rn(y(y+γ+δ+1);α,β,γ,δ).

§18.28(xi) Limits for q1

Bannai and Ito (1984) introduced OP’s, called the Bannai–Ito polynomials which are the limit for q1 of the q-Racah polynomials. They have to be included in the classification by Leonard (1982), mentioned in §18.28(viii). In Tsujimoto et al. (2012) an extension of the Bannai–Ito polynomials occurs as eigenfunctions of a Dunkl type operator. Genest et al. (2016) showed that these polynomials coincide with the nonsymmetric Wilson polynomials in Groenevelt (2007).