§18.33 Polynomials Orthogonal on the Unit Circle

§18.33(i) Definition

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

 18.33.1 $\frac{1}{2\pi\mathrm{i}}\int_{|z|=1}\phi_{n}(z)\overline{\phi_{m}(z)}w(z)\frac% {\,\mathrm{d}z}{z}=\delta_{n,m},$

where the bar signifies complex conjugate. Simon (2005a, b) gives the general theory of these OP’s in terms of monic OP’s $\Phi_{n}(x)$, see §18.33(vi).

§18.33(ii) Recurrence Relations

Denote

 18.33.2 $\phi_{n}(z)=\kappa_{n}z^{n}+\sum_{\ell=1}^{n}\kappa_{n,n-\ell}z^{n-\ell},$ ⓘ Symbols: $z$: complex variable, $\ell$: nonnegative integer, $n$: nonnegative integer and $\phi_{n}(z)$: polynomials Permalink: http://dlmf.nist.gov/18.33.E2 Encodings: TeX, pMML, png See also: Annotations for §18.33(ii), §18.33 and Ch.18

where $\kappa_{n}(>0)$, and $\kappa_{n,n-\ell}(\in\mathbb{C})$ are constants. Also denote

 18.33.3 $\phi_{n}^{*}(z)=z^{n}\overline{\phi_{n}({\overline{z}}^{-1})}={\kappa_{n}}+% \sum_{\ell=1}^{n}\overline{\kappa}_{n,n-\ell}z^{\ell},$ ⓘ Symbols: $\overline{\NVar{z}}$: complex conjugate, $z$: complex variable, $\ell$: nonnegative integer, $n$: nonnegative integer and $\phi_{n}(z)$: polynomials Referenced by: Erratum (V1.0.10) for Equation (18.33.3) Permalink: http://dlmf.nist.gov/18.33.E3 Encodings: TeX, pMML, png Errata (effective with 1.0.10): Originally this equation was written incorrectly as $\phi_{n}^{*}(z)={\kappa_{n}}z^{n}+\sum_{\ell=1}^{n}\overline{\kappa}_{n,n-\ell% }z^{n-\ell}$. Also, the equality $\phi_{n}^{*}(z)=z^{n}\overline{\phi_{n}({\overline{z}}^{-1})}$ has been added. Reported 2014-11-10 by Roderick Wong See also: Annotations for §18.33(ii), §18.33 and Ch.18

where the bar again signifies complex conjugate. Then

 18.33.4 $\displaystyle\kappa_{n}z\phi_{n}(z)$ $\displaystyle=\kappa_{n+1}\phi_{n+1}(z)-\phi_{n+1}(0)\phi_{n+1}^{*}(z),$ ⓘ Symbols: $z$: complex variable, $n$: nonnegative integer and $\phi_{n}(z)$: polynomials Referenced by: §18.33(ii) Permalink: http://dlmf.nist.gov/18.33.E4 Encodings: TeX, pMML, png See also: Annotations for §18.33(ii), §18.33 and Ch.18 18.33.5 $\displaystyle\kappa_{n}\phi_{n+1}(z)$ $\displaystyle=\kappa_{n+1}z\phi_{n}(z)+\phi_{n+1}(0)\phi_{n}^{*}(z),$ ⓘ Symbols: $z$: complex variable, $n$: nonnegative integer and $\phi_{n}(z)$: polynomials Referenced by: §18.33(ii) Permalink: http://dlmf.nist.gov/18.33.E5 Encodings: TeX, pMML, png See also: Annotations for §18.33(ii), §18.33 and Ch.18 18.33.6 $\displaystyle\kappa_{n}\phi_{n}(0)\phi_{n+1}(z)+\kappa_{n-1}\phi_{n+1}(0)z\phi% _{n-1}(z)$ $\displaystyle=\left(\kappa_{n}\phi_{n+1}(0)+\kappa_{n+1}\phi_{n}(0)z\right)% \phi_{n}(z).$ ⓘ Symbols: $z$: complex variable, $n$: nonnegative integer and $\phi_{n}(z)$: polynomials Referenced by: §18.33(ii) Permalink: http://dlmf.nist.gov/18.33.E6 Encodings: TeX, pMML, png See also: Annotations for §18.33(ii), §18.33 and Ch.18

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 $w({\mathrm{e}}^{\mathrm{i}\phi})=w({\mathrm{e}}^{-\mathrm{i}\phi})$. Set

 18.33.7 $\displaystyle w_{1}(x)$ $\displaystyle=(1-x^{2})^{-\frac{1}{2}}w\left(x+\mathrm{i}(1-x^{2})^{\frac{1}{2% }}\right),$ $\displaystyle w_{2}(x)$ $\displaystyle=(1-x^{2})^{\frac{1}{2}}w\left(x+\mathrm{i}(1-x^{2})^{\frac{1}{2}% }\right).$ ⓘ Symbols: $\mathrm{i}$: imaginary unit, $w(x)$: weight function, $w_{x}$: weights and $x$: real variable Permalink: http://dlmf.nist.gov/18.33.E7 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §18.33(iii), §18.33 and Ch.18

Let $\{p_{n}(x)\}$ and $\{q_{n}(x)\}$, $n=0,1,\dots$, be OP’s with weight functions $w_{1}(x)$ and $w_{2}(x)$, respectively, on $(-1,1)$. Then

 18.33.8 $\displaystyle p_{n}\left(\tfrac{1}{2}(z+z^{-1})\right)$ $\displaystyle=\text{(const.)}\times\left(z^{-n}\phi_{2n}(z)+z^{n}\phi_{2n}(z^{% -1})\right)=\text{(const.)}\times\left(z^{-n+1}\phi_{2n-1}(z)+z^{n-1}\phi_{2n-% 1}(z^{-1})\right),$ ⓘ Symbols: $z$: complex variable, $n$: nonnegative integer, $p_{n}(x)$: an orthogonal polynomial and $\phi_{n}(z)$: polynomials Referenced by: §18.33(iii) Permalink: http://dlmf.nist.gov/18.33.E8 Encodings: TeX, pMML, png See also: Annotations for §18.33(iii), §18.33 and Ch.18 18.33.9 $\displaystyle q_{n}\left(\tfrac{1}{2}(z+z^{-1})\right)$ $\displaystyle=\text{(const.)}\times\frac{z^{-n-1}\phi_{2n+2}(z)-z^{n+1}\phi_{2% n+2}(z^{-1})}{z-z^{-1}}=\text{(const.)}\times\frac{z^{-n}\phi_{2n+1}(z)-z^{n}% \phi_{2n+1}(z^{-1})}{z-z^{-1}}.$ ⓘ Symbols: $z$: complex variable, $n$: nonnegative integer, $q_{n}(x)$: an orthogonal polynomial and $\phi_{n}(z)$: polynomials Referenced by: §18.33(iii), §18.33(iii) Permalink: http://dlmf.nist.gov/18.33.E9 Encodings: TeX, pMML, png See also: Annotations for §18.33(iii), §18.33 and Ch.18

Conversely,

 18.33.10 $\displaystyle z^{-n}\phi_{2n}(z)$ $\displaystyle={A_{n}p_{n}\left(\tfrac{1}{2}(z+z^{-1})\right)+B_{n}(z-z^{-1})q_% {n-1}\left(\tfrac{1}{2}(z+z^{-1})\right)},$ 18.33.11 $\displaystyle z^{-n+1}\phi_{2n-1}(z)$ $\displaystyle={C_{n}p_{n}\left(\tfrac{1}{2}(z+z^{-1})\right)+D_{n}(z-z^{-1})q_% {n-1}\left(\tfrac{1}{2}(z+z^{-1})\right)},$

where $A_{n}$, $B_{n}$, $C_{n}$, and $D_{n}$ are independent of $z$.

Instead of (18.33.9) one might take monic OP’s $\{q_{n}(x)\}$ with weight function $(1+x)w_{1}(x)$, and then express $q_{n}\left(\tfrac{1}{2}(z+z^{-1})\right)$ in terms of $\phi_{2n}(z^{\pm 1})$ or $\phi_{2n+1}(z^{\pm 1})$. After a quadratic transformation (18.2.23) this would express OP’s on $[-1,1]$ with an even orthogonality measure in terms of the $\phi_{n}$. See Zhedanov (1998, §2).

§18.33(iv) Special Cases

Trivial

 18.33.12 $\displaystyle\phi_{n}(z)$ $\displaystyle=z^{n},$ $\displaystyle w(z)$ $\displaystyle=1.$ ⓘ Symbols: $w(x)$: weight function, $z$: complex variable, $n$: nonnegative integer and $\phi_{n}(z)$: polynomials Permalink: http://dlmf.nist.gov/18.33.E12 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §18.33(iv), §18.33(iv), §18.33 and Ch.18

 18.33.13 $\phi_{n}(z)=\sum_{\ell=0}^{n}\frac{{\left(\lambda+1\right)_{\ell}}{\left(% \lambda\right)_{n-\ell}}}{\ell!\,(n-\ell)!}\,z^{\ell}=\frac{{\left(\lambda% \right)_{n}}}{n!}{{}_{2}F_{1}}\left({-n,\lambda+1\atop-\lambda-n+1};z\right),$

with

 18.33.14 $\displaystyle w(z)$ $\displaystyle=\left(1-\tfrac{1}{2}(z+z^{-1})\right)^{\lambda},$ $\displaystyle w_{1}(x)$ $\displaystyle=(1-x)^{\lambda-\frac{1}{2}}(1+x)^{-\frac{1}{2}},$ $\displaystyle w_{2}(x)$ $\displaystyle=(1-x)^{\lambda+\frac{1}{2}}(1+x)^{\frac{1}{2}},$ $\lambda>-\tfrac{1}{2}$. ⓘ Symbols: $w(x)$: weight function, $w_{x}$: weights, $z$: complex variable and $x$: real variable Referenced by: §18.33(v) Permalink: http://dlmf.nist.gov/18.33.E14 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §18.33(iv), §18.33(iv), §18.33 and Ch.18

For the hypergeometric function ${{}_{2}F_{1}}$ 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 $\phi_{n}(z)=\sum_{\ell=0}^{n}\frac{\left(aq^{2};q^{2}\right)_{\ell}\left(a;q^{% 2}\right)_{n-\ell}}{\left(q^{2};q^{2}\right)_{\ell}\left(q^{2};q^{2}\right)_{n% -\ell}}(q^{-1}z)^{\ell}=\frac{\left(a;q^{2}\right)_{n}}{\left(q^{2};q^{2}% \right)_{n}}{{}_{2}\phi_{1}}\left({aq^{2},q^{-2n}\atop a^{-1}q^{2-2n}};q^{2},% \frac{qz}{a}\right),$

with

 18.33.16 $w(z)={\left|\left(qz;q^{2}\right)_{\infty}\Bigm{/}\left(aqz;q^{2}\right)_{% \infty}\right|}^{2},$ $a^{2}q^{2}<1$.

For the notation, including the basic hypergeometric function ${{}_{2}\phi_{1}}$, 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 $\{\phi_{n}(z)\}$ Simon (2005a, b) uses monic polynomials $\Phi_{n}(z)$. Let $\mu$ be a probability measure on the unit circle of which the support is an infinite set. A system of monic polynomials $\{\Phi_{n}(z)\}$, $n=0,1,\dots$, where $\Phi_{n}(x)$ is of proper degree $n$, is orthogonal on the unit circle with respect to the measure $\mu$ if

 18.33.17 $\int_{|z|=1}\Phi_{n}(z)\overline{\Phi_{m}(z)}\,\mathrm{d}\mu(z)=\kappa_{n}^{-2% }\delta_{n,m},$

where the bar signifies complex conjugate and $\kappa_{n}>0$, $\kappa_{0}=1$. Then the corresponding orthonormal polynomials are

 18.33.18 $\phi_{n}(z)=\kappa_{n}\Phi_{n}(z).$ ⓘ Symbols: $z$: complex variable, $n$: nonnegative integer and $\phi_{n}(z)$: polynomials Source: Simon (2005a, (1.1.2), (1.1.3)) Permalink: http://dlmf.nist.gov/18.33.E18 Encodings: TeX, pMML, png See also: Annotations for §18.33(vi), §18.33 and Ch.18

If the measure $\mu$ is absolutely continuous, i.e.,

 18.33.19 $\,\mathrm{d}\mu(z)=\frac{1}{2\pi\mathrm{i}}\,w(z)\frac{\,\mathrm{d}z}{z}$

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

 18.33.20 $\frac{1}{2\pi\mathrm{i}}\int_{|z|=1}\Phi_{n}(z)\overline{\Phi_{m}(z)}w(z)\frac% {\,\mathrm{d}z}{z}=\kappa_{n}^{-2}\delta_{n,m}.$

Recurrence Relations

For a polynomial

 18.33.21 $p(z)=\sum_{k=0}^{n}c_{k}z^{k},$ $c_{n}\neq 0$, ⓘ Symbols: $z$: complex variable, $k$: nonnegative integer and $n$: nonnegative integer Source: Simon (2005a, (1.1.7)) Permalink: http://dlmf.nist.gov/18.33.E21 Encodings: TeX, pMML, png See also: Annotations for §18.33(vi), §18.33(vi), §18.33 and Ch.18

with complex coefficients $c_{k}$ and of a certain degree $n$ define the reversed polynomial $p^{*}(z)$ by

 18.33.22 $p^{*}(z)\equiv z^{n}\overline{p({\overline{z}}^{-1})}=\sum_{k=0}^{n}\overline{% c_{n-k}}z^{k}.$ ⓘ Symbols: $\overline{\NVar{z}}$: complex conjugate, $\equiv$: equals by definition, $z$: complex variable, $k$: nonnegative integer and $n$: nonnegative integer Source: Simon (2005a, (1.1.7)) Permalink: http://dlmf.nist.gov/18.33.E22 Encodings: TeX, pMML, png See also: Annotations for §18.33(vi), §18.33(vi), §18.33 and Ch.18

The Verblunsky coefficients (also called Schur parameters or reflection coefficients) are the coefficients $\alpha_{n}$ in the Szegő recurrence relations

 18.33.23 $\displaystyle\Phi_{n+1}(z)$ $\displaystyle=z\Phi_{n}(z)-\overline{\alpha_{n}}\,\Phi_{n}^{*}(z),$ ⓘ Symbols: $\overline{\NVar{z}}$: complex conjugate, $z$: complex variable and $n$: nonnegative integer Proved: Simon (2005a, Theorem 1.5.2)(proved) Referenced by: §18.33(vi), §18.33(vi), §18.33(vi), §18.33(vi) Permalink: http://dlmf.nist.gov/18.33.E23 Encodings: TeX, pMML, png See also: Annotations for §18.33(vi), §18.33(vi), §18.33 and Ch.18 18.33.24 $\displaystyle\Phi_{n+1}^{*}(z)$ $\displaystyle=\Phi_{n}^{*}(z)-\alpha_{n}z\Phi_{n}(z).$ ⓘ Symbols: $z$: complex variable and $n$: nonnegative integer Proved: Simon (2005a, Theorem 1.5.2)(proved) Referenced by: §18.33(vi), §18.33(vi), §18.33(vi), §18.33(vi) Permalink: http://dlmf.nist.gov/18.33.E24 Encodings: TeX, pMML, png See also: Annotations for §18.33(vi), §18.33(vi), §18.33 and Ch.18

Then

 18.33.25 $\displaystyle\alpha_{n}$ $\displaystyle=-\overline{\Phi_{n+1}(0)},$ $\displaystyle|\alpha_{n}|$ $\displaystyle<1,$ ⓘ Symbols: $\overline{\NVar{z}}$: complex conjugate and $n$: nonnegative integer Proved: Simon (2005a, (1.5.20))(proved) Referenced by: §18.33(vi) Permalink: http://dlmf.nist.gov/18.33.E25 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §18.33(vi), §18.33(vi), §18.33 and Ch.18 18.33.26 $\displaystyle\rho_{n}$ $\displaystyle\equiv\sqrt{1-|\alpha_{n}|^{2}}=\frac{\kappa_{n}}{\kappa_{n+1}}.$ ⓘ Symbols: $\equiv$: equals by definition and $n$: nonnegative integer Proved: Simon (2005a, (1.5.12))(proved) Source: Simon (2005a, (1.5.20)) Permalink: http://dlmf.nist.gov/18.33.E26 Encodings: TeX, pMML, png See also: Annotations for §18.33(vi), §18.33(vi), §18.33 and Ch.18

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

 18.33.27 $\displaystyle z\Phi_{n}(z)$ $\displaystyle=\rho_{n}^{-2}\left(\Phi_{n+1}(z)+\overline{\alpha_{n}}\Phi_{n+1}% ^{*}(z)\right),$ ⓘ Symbols: $\overline{\NVar{z}}$: complex conjugate, $z$: complex variable and $n$: nonnegative integer Proved: Simon (2005a, Theorem 1.5.4)(proved) Referenced by: §18.33(vi) Permalink: http://dlmf.nist.gov/18.33.E27 Encodings: TeX, pMML, png See also: Annotations for §18.33(vi), §18.33(vi), §18.33 and Ch.18 18.33.28 $\displaystyle\Phi_{n}^{*}(z)$ $\displaystyle=\rho_{n}^{-2}\left(\Phi_{n+1}^{*}(z)+\alpha_{n}\Phi_{n+1}(z)% \right).$ ⓘ Symbols: $z$: complex variable and $n$: nonnegative integer Proved: Simon (2005a, Theorem 1.5.4)(proved) Permalink: http://dlmf.nist.gov/18.33.E28 Encodings: TeX, pMML, png See also: Annotations for §18.33(vi), §18.33(vi), §18.33 and Ch.18

Combination of (18.33.23) and (18.33.24) gives

 18.33.29 $\Phi_{n+1}(z)\pm\Phi_{n+1}^{*}(z)=(1\mp\alpha_{n})z\Phi_{n}(z)+(\pm 1-% \overline{\alpha_{n}})\Phi_{n}^{*}(z),$ ⓘ Symbols: $\overline{\NVar{z}}$: complex conjugate, $z$: complex variable and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/18.33.E29 Encodings: TeX, pMML, png See also: Annotations for §18.33(vi), §18.33(vi), §18.33 and Ch.18

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

 18.33.30 $\overline{\alpha_{n-1}}\,\Phi_{n+1}(z)=\left(\overline{\alpha_{n}}+\overline{% \alpha_{n-1}}\,z\right)\Phi_{n}(z)-\overline{\alpha_{n}}\rho_{n-1}^{2}z\Phi_{n% -1}(z)$ ⓘ Symbols: $\overline{\NVar{z}}$: complex conjugate, $z$: complex variable and $n$: nonnegative integer Proved: Simon (2005a, (1.5.47))(proved) Permalink: http://dlmf.nist.gov/18.33.E30 Encodings: TeX, pMML, png See also: Annotations for §18.33(vi), §18.33(vi), §18.33 and Ch.18

for $n>0$, while $\Phi_{1}(z)=z-\overline{\alpha_{0}}$.

Verblunsky’s Theorem

This states that for any sequence $\{\alpha_{n}\}_{n=0}^{\infty}$ with $\alpha_{n}\in\mathbb{C}$ and $|\alpha_{n}|<1$ the polynomials $\Phi_{n}(z)$ generated by the recurrence relations (18.33.23), (18.33.24) with $\Phi_{0}(z)=1$ satisfy the orthogonality relation (18.33.17) for a unique probability measure $\mu$ 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 $\mu$ in (18.33.17)) and with $\alpha_{n}$ the Verblunsky coefficients in (18.33.23), (18.33.24), Szegő’s theorem states that

 18.33.31 $\prod_{j=0}^{\infty}\left(1-\left|\alpha_{j}\right|^{2}\right)=\exp\left(\frac% {1}{2\pi\mathrm{i}}\int_{|z|=1}\ln\left(w(z)\right)\frac{\,\mathrm{d}z}{z}% \right).$

By (18.33.25) $|\alpha_{j}|<1$, so the infinite product in (18.33.31) converges, although the limit may be zero. In particular, by (18.33.31),

 18.33.32 $\sum_{j=0}^{\infty}|\alpha_{j}|^{2}<\infty\quad\Longleftrightarrow\quad\frac{1% }{2\pi\mathrm{i}}\int_{|z|=1}\ln\left((w(z)\right)\frac{\,\mathrm{d}z}{z}>-\infty.$