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§18.33 Polynomials Orthogonal on the Unit Circle

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§18.33(i) Definition

A system of polynomials {ϕn(z)}, n=0,1,, where ϕn(z) is of proper degree n, is orthonormal on the unit circle with respect to the weight function w(z) (0) if

18.33.1 12πi|z|=1ϕn(z)ϕm(z)¯w(z)dzz=δn,m,

where the bar signifies complex conjugate. See Simon (2005a, b) for general theory.

§18.33(ii) Recurrence Relations

Denote

18.33.2 ϕn(z)=κnzn+=1nκn,n-zn-,

where κn(>0), and κn,n-() are constants. Also denote

18.33.3 ϕn*(z)=κnzn+=1nκ¯n,n-zn-,

where the bar again signifies compex conjugate. Then

18.33.4 κnzϕn(z) =κn+1ϕn+1(z)-ϕn+1(0)ϕn+1*(z),
18.33.5 κnϕn+1(z) =κn+1zϕn(z)+ϕn+1(0)ϕn*(z),
18.33.6 κnϕn(0)ϕn+1(z)+κn-1ϕn+1(0)zϕn-1(z) =(κnϕn+1(0)+κn+1ϕn(0)z)ϕn(z).

§18.33(iii) Connection with OP’s on the Line

Assume that w(eiϕ)=w(e-iϕ). Set

18.33.7 w1(x) =(1-x2)-12w(x+i(1-x2)12),
w2(x) =(1-x2)12w(x+i(1-x2)12).

Let {pn(x)} and {qn(x)}, n=0,1,, be OP’s with weight functions w1(x) and w2(x), respectively, on (-1,1). Then

18.33.8 pn(12(z+z-1)) =(const.)×(z-nϕ2n(z)+znϕ2n(z-1))
=(const.)×(z-n+1ϕ2n-1(z)+zn-1ϕ2n-1(z-1)),
18.33.9 qn(12(z+z-1)) =(const.)×z-n-1ϕ2n+2(z)-zn+1ϕ2n+2(z-1)z-z-1
=(const.)×z-nϕ2n+1(z)-znϕ2n+1(z-1)z-z-1.

Conversely,

18.33.10 z-nϕ2n(z) =Anpn(12(z+z-1))+Bn(z-z-1)qn-1(12(z+z-1)),
18.33.11 z-n+1ϕ2n-1(z) =Cnpn(12(z+z-1))+Dn(z-z-1)qn-1(12(z+z-1)),

where An, Bn, Cn, and Dn are independent of z.

§18.33(iv) Special Cases

Trivial

Szegö–Askey

18.33.13 ϕn(z)==0n(λ+1)(λ)n-!(n-)!z=(λ)nn!F12(-n,λ+1-λ-n+1;z),

with

18.33.14 w(z) =(1-12(z+z-1))λ,
w1(x) =(1-x)λ-12(1+x)-12,
w2(x) =(1-x)λ+12(1+x)12,
λ>-12.

For the hypergeometric function F12 see §§15.1 and 15.2(i).

Askey

18.33.15 ϕn(z)==0n(aq2;q2)(a;q2)n-(q2;q2)(q2;q2)n-(q-1z)=(a;q2)n(q2;q2)nϕ12(aq2,q-2na-1q2-2n;q2,qza),

with

18.33.16 w(z)=|(qz;q2)/(aqz;q2)|2,
a2q2<1.

For the notation, including the basic hypergeometric function ϕ12, see §§17.2 and 17.4(i).

When a=0 the Askey case is also known as the Rogers–Szegö case.

§18.33(v) Biorthogonal Polynomials on the Unit Circle

See Baxter (1961) for general theory. See Askey (1982) and Pastro (1985) for special cases extending (18.33.13)–(18.33.14) and (18.33.15)–(18.33.16), respectively. See Gasper (1981) and Hendriksen and van Rossum (1986) for relations with Laurent polynomials orthogonal on the unit circle. See Al-Salam and Ismail (1994) for special biorthogonal rational functions on the unit circle.