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§18.33 Polynomials Orthogonal on the Unit Circle

  1. §18.33(i) Definition
  2. §18.33(ii) Recurrence Relations
  3. §18.33(iii) Connection with OP’s on the Line
  4. §18.33(iv) Special Cases
  5. §18.33(v) Biorthogonal Polynomials on the Unit Circle
  6. §18.33(vi) Alternative Set-up with Monic Polynomials

§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. Simon (2005a, b) gives the general theory of these OP’s in terms of monic OP’s Φn(x), see §18.33(vi).

§18.33(ii) Recurrence Relations


18.33.2 ϕn(z)=κnzn+=1nκn,nzn,

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

18.33.3 ϕn(z)=znϕn(z¯1)¯=κn+=1nκ¯n,nz,

where the bar again signifies complex 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)+κn1ϕn+1(0)zϕn1(z) =(κnϕn+1(0)+κn+1ϕn(0)z)ϕn(z).

For an alternative and more detailed approach to the recurrence relations, see §18.33(vi).

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

Assume that w(eiϕ)=w(eiϕ). Set

18.33.7 w1(x) =(1x2)12w(x+i(1x2)12),
w2(x) =(1x2)12w(x+i(1x2)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+z1)) =(const.)×(znϕ2n(z)+znϕ2n(z1))=(const.)×(zn+1ϕ2n1(z)+zn1ϕ2n1(z1)),
18.33.9 qn(12(z+z1)) =(const.)×zn1ϕ2n+2(z)zn+1ϕ2n+2(z1)zz1=(const.)×znϕ2n+1(z)znϕ2n+1(z1)zz1.


18.33.10 znϕ2n(z) =Anpn(12(z+z1))+Bn(zz1)qn1(12(z+z1)),
18.33.11 zn+1ϕ2n1(z) =Cnpn(12(z+z1))+Dn(zz1)qn1(12(z+z1)),

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

Instead of (18.33.9) one might take monic OP’s {qn(x)} with weight function (1+x)w1(x), and then express qn(12(z+z1)) in terms of ϕ2n(z±1) or ϕ2n+1(z±1). After a quadratic transformation (18.2.23) this would express OP’s on [1,1] with an even orthogonality measure in terms of the ϕn. See Zhedanov (1998, §2).

§18.33(iv) Special Cases


18.33.12 ϕn(z) =zn,
w(z) =1.


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


18.33.14 w(z) =(112(z+z1))λ,
w1(x) =(1x)λ12(1+x)12,
w2(x) =(1x)λ+12(1+x)12,

For the hypergeometric function F12 see §§15.1 and 15.2(i). Askey (1982a) and Sri Ranga (2010) give more general results leading to what seem to be the right analogues of Jacobi polynomials on the unit circle.


18.33.15 ϕn(z)==0n(aq2;q2)(a;q2)n(q2;q2)(q2;q2)n(q1z)=(a;q2)n(q2;q2)nϕ12(aq2,q2na1q22n;q2,qza),


18.33.16 w(z)=|(qz;q2)/(aqz;q2)|2,

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–Szegő case. See for a more general class Costa et al. (2012).

§18.33(v) Biorthogonal Polynomials on the Unit Circle

See Baxter (1961) for general theory. See Askey (1982a) 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.

§18.33(vi) Alternative Set-up with Monic Polynomials

Instead of orthonormal polynomials {ϕn(z)} Simon (2005a, b) uses monic polynomials Φn(z). Let μ be a probability measure on the unit circle of which the support is an infinite set. A system of monic polynomials {Φn(z)}, n=0,1,, where Φn(x) is of proper degree n, is orthogonal on the unit circle with respect to the measure μ if

18.33.17 |z|=1Φn(z)Φm(z)¯dμ(z)=κn2δn,m,

where the bar signifies complex conjugate and κn>0, κ0=1. Then the corresponding orthonormal polynomials are

18.33.18 ϕn(z)=κnΦn(z).

If the measure μ is absolutely continuous, i.e.,

18.33.19 dμ(z)=12πiw(z)dzz

for some weight function w(z) (0) then (18.33.17) (see also (18.33.1)) takes the form

18.33.20 12πi|z|=1Φn(z)Φm(z)¯w(z)dzz=κn2δn,m.

Recurrence Relations

For a polynomial

18.33.21 p(z)=k=0nckzk,

with complex coefficients ck and of a certain degree n define the reversed polynomial p(z) by

18.33.22 p(z)znp(z¯1)¯=k=0ncnk¯zk.

The Verblunsky coefficients (also called Schur parameters or reflection coefficients) are the coefficients αn in the Szegő recurrence relations

18.33.23 Φn+1(z) =zΦn(z)αn¯Φn(z),
18.33.24 Φn+1(z) =Φn(z)αnzΦn(z).


18.33.25 αn =Φn+1(0)¯,
|αn| <1,
18.33.26 ρn 1|αn|2=κnκn+1.

Equivalent to the recurrence relations (18.33.23), (18.33.24) are the inverse Szegő recurrence relations

18.33.27 zΦn(z) =ρn2(Φn+1(z)+αn¯Φn+1(z)),
18.33.28 Φn(z) =ρn2(Φn+1(z)+αnΦn+1(z)).

Combination of (18.33.23) and (18.33.24) gives

18.33.29 Φn+1(z)±Φn+1(z)=(1αn)zΦn(z)+(±1αn¯)Φn(z),

while combination of (18.33.27) and (18.33.23) gives the three-term recurrence relation

18.33.30 αn1¯Φn+1(z)=(αn¯+αn1¯z)Φn(z)αn¯ρn12zΦn1(z)

for n>0, while Φ1(z)=zα0¯.

Verblunsky’s Theorem

This states that for any sequence {αn}n=0 with αn and |αn|<1 the polynomials Φn(z) generated by the recurrence relations (18.33.23), (18.33.24) with Φ0(z)=1 satisfy the orthogonality relation (18.33.17) for a unique probability measure μ with infinite support on the unit circle. See Simon (2005a, p. 2, item (2)).

Szegő’s Theorem

For w(z) as in (18.33.19) (or more generally as the weight function of the absolutely continuous part of the measure μ in (18.33.17)) and with αn the Verblunsky coefficients in (18.33.23), (18.33.24), Szegő’s theorem states that

18.33.31 j=0(1|αj|2)=exp(12πi|z|=1ln(w(z))dzz).

By (18.33.25) |αj|<1, so the infinite product in (18.33.31) converges, although the limit may be zero. In particular, by (18.33.31),

18.33.32 j=0|αj|2<12πi|z|=1ln((w(z))dzz>.