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§18.28 Askey–Wilson Class

Contents

§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 {q-y+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+z-1)) in the Askey–Wilson case, and Pn(y)=pn(q-y+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 (2005, 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(cosθ)=pn(cosθ;a,b,c,d| q)=a-n=0nq(abq,acq,adq;q)n-(q-n,abcdqn-1;q)(q;q)×j=0-1(1-2aqjcosθ+a2q2j)=a-n(ab,ac,ad;q)nϕ34(q-n,abcdqn-1,aθ,a-θab,ac,ad;q,q).

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. Furthermore, |ab|, |ac|, |ad|, |bc|, |bd|, |cd|<1. Then

18.28.2 -11pn(x)pm(x)w(x)x=hnδn,m,
|a|,|b|,|c|,|d|1,

where

18.28.3 2πsinθw(cosθ)=|(2θ;q)(aθ,bθ,cθ,dθ;q)|2,
18.28.4 h0=(abcd;q)(q,ab,ac,ad,bc,bd,cd;q),
18.28.5 hn=h0(1-abcdqn-1)(q,ab,ac,ad,bc,bd,cd;q)n(1-abcdq2n-1)(abcd;q)n,
n=1,2,.

More generally, without the constraints in (18.28.2),

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

with w(x) and hn as above. Also, x are the points 12(αq+α-1q-) 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.

§18.28(iii) Al-Salam–Chihara Polynomials

18.28.7 Qn(cosθ;a,b| q)=pn(cosθ;a,b,0,0| q)=a-n=0nq(abq;q)n-(q-n;q)(q;q)j=0-1(1-2aqjcosθ+a2q2j)=(ab;q)nanϕ23(q-n,aθ,a-θab,0;q,q)=(b-θ;q)nnθϕ12(q-n,aθb-1q1-nθ;q,b-1q-θ).
18.28.8 12π0πQn(cosθ;a,b| q)Qm(cosθ;a,b| q)|(2θ;q)(aθ,bθ;q)|2θ=δn,m(qn+1,abqn;q),
a,b or a=b¯; |ab|<1; |a|,|b|1.

More generally, without the constraints |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) q-1-Al-Salam–Chihara Polynomials

18.28.9 Qn(12(aq-y+a-1qy);a,b| q-1)=(-1)nbnq-12n(n-1)((ab)-1;q)nϕ13(q-n,q-y,a-2qy(ab)-1;q,qnab-1).
18.28.10 y=0(1-q2ya-2)(a-2,(ab)-1;q)y(1-a-2)(q,bqa-1;q)y(ba-1)y×qy2Qn(12(aq-y+a-1qy);a,b| q-1)×Qm(12(aq-y+a-1qy);a,b| q-1)=(qa-2;q)(ba-1q;q)(q,(ab)-1;q)n(ab)nq-n2δn,m.

Eq. (18.28.10) is valid when either

18.28.11 0<q<1,a,b,ab>1,a-1b<q-1,

or

18.28.12 0<q<1,a/,b/,(a)(b)>0,a-1b<q-1.

If, in addition to (18.28.11) or (18.28.12), we have a-1bq, then the measure in (18.28.10) is uniquely determined. Also, if q<a-1b<q-1, 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)n-(n-2)θ
=(β;q)n(q;q)nnθϕ12(q-n,ββ-1q1-n;q,β-1q-2θ).
18.28.14 Cn(cosθ;β| q) =(β2;q)n(q;q)nβ12nϕ34(q-n,β2qn,β12θ,β12-θβq12,-β,-βq12;q,q).
18.28.15 12π0πCn(cosθ;β| q)Cm(cosθ;β| q)|(2θ;q)(β2θ;q)|2θ=(β,βq;q)(β2,q;q)(1-β)(β2;q)n(1-βqn)(q;q)nδn,m,
-1<β<1.

These polynomials are also called Rogers polynomials.

§18.28(vi) Continuous q-Hermite Polynomials

18.28.16 Hn(cosθ| q)==0n(q;q)n(n-2)θ(q;q)(q;q)n-=nθϕ02(q-n,0-;q,qn-2θ).
18.28.17 12π0πHn(cosθ| q)Hm(cosθ| q)|(2θ;q)|2θ=δn,m(qn+1;q).

§18.28(vii) Continuous q-1-Hermite Polynomials

18.28.18 hn(sinht| q)==0nq12(+1)(q-n;q)(q;q)(n-2)t=ntϕ11(q-n0;q,-q-2t)=-nHn(sinht| q-1).

For continuous q-1-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=q-y+γδqy+1,

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

For ωy and hn see Koekoek et al. (2010, Eq. (14.2.2)).