18 Orthogonal Polynomials Other Orthogonal Polynomials 18.32 OP’s with Respect to Freud Weights 18.34 Bessel Polynomials

§18.33 Polynomials Orthogonal on the Unit Circle

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Keywords:: polynomials orthogonal on the unit circle
Referenced by:: Erratum (V1.2.0) §18.33
Permalink:: http://dlmf.nist.gov/18.33
See also:: Annotations for Ch.18

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

§18.33(i) Definition

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Keywords:: definition, polynomials orthogonal on the unit circle
Notes:: See Szegő (1975, (11.1.8)).
Permalink:: http://dlmf.nist.gov/18.33.i
Addition (effective with 1.2.0):: A sentence was added at the end of this subsection with references.
See also:: Annotations for §18.33 and Ch.18

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		$\frac{1}{2\pi\mathrm{i}}\int_{\|z\|=1}\phi_{n}(z)\overline{\phi_{m}(z)}w(z)\frac% {\,\mathrm{d}z}{z}=\delta_{n,m},$
ⓘ Symbols: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\pi$ : the ratio of the circumference of a circle to its diameter, $\overline{\NVar{z}}$ : complex conjugate, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $w(x)$ : weight function, $z$ : complex variable, $m$ : nonnegative integer, $n$ : nonnegative integer and $\phi_{n}(z)$ : polynomials Referenced by: §18.33(vi) Permalink: http://dlmf.nist.gov/18.33.E1 Encodings: TeX, pMML, png See also: Annotations for §18.33(i), §18.33 and Ch.18

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

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Keywords:: polynomials orthogonal on the unit circle, recurrence relations
Notes:: See Szegő (1975, (11.4.6), (11.4.7)). (18.33.6) follows from (18.33.4), (18.33.5).
Permalink:: http://dlmf.nist.gov/18.33.ii
Addition (effective with 1.2.0):: A sentence was added at the end of this subsection.
See also:: Annotations for §18.33 and Ch.18

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

ⓘ

Keywords:: connection with orthogonal polynomials on the line, polynomials orthogonal on the unit circle
Notes:: See Szegő (1975, (11.5.2)). (18.33.10) and (18.33.11) follow from (18.33.8), (18.33.9).
Permalink:: http://dlmf.nist.gov/18.33.iii
Addition (effective with 1.2.0):: A paragraph was added at the end of this subsection.
See also:: Annotations for §18.33 and Ch.18

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

18.33.7	$\displaystyle w_{1}(x)$	$\displaystyle=(1-x^{2})^{-\frac{1}{2}}w\left(x+\mathrm{i}(1-x^{2})^{\frac{1}{2% }}\right),$
18.33.7	$\displaystyle w_{2}(x)$	$\displaystyle=(1-x^{2})^{\frac{1}{2}}w\left(x+\mathrm{i}(1-x^{2})^{\frac{1}{2}% }\right).$
ⓘ 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	$\displaystyle z^{-n}\phi_{2n}(z)$	$\displaystyle={A_{n}p_{n}\left(\tfrac{1}{2}(z+z^{-1})\right)+B_{n}(z-z^{-1})q_% {n-1}\left(\tfrac{1}{2}(z+z^{-1})\right)},$
ⓘ Symbols: $z$ : complex variable, $n$ : nonnegative integer, $q_{n}(x)$ : an orthogonal polynomial, $p_{n}(x)$ : an orthogonal polynomial, $A_{n}$ : coefficient, $B_{n}$ : coefficient and $\phi_{n}(z)$ : polynomials Referenced by: §18.33(iii) Permalink: http://dlmf.nist.gov/18.33.E10 Encodings: TeX, pMML, png See also: Annotations for §18.33(iii), §18.33 and Ch.18
18.33.11	$\displaystyle z^{-n+1}\phi_{2n-1}(z)$	$\displaystyle={C_{n}p_{n}\left(\tfrac{1}{2}(z+z^{-1})\right)+D_{n}(z-z^{-1})q_% {n-1}\left(\tfrac{1}{2}(z+z^{-1})\right)},$
ⓘ Symbols: $z$ : complex variable, $n$ : nonnegative integer, $q_{n}(x)$ : an orthogonal polynomial, $p_{n}(x)$ : an orthogonal polynomial, $C_{n}$ : coefficient, $D_{n}$ : coefficient and $\phi_{n}(z)$ : polynomials Referenced by: §18.33(iii) Permalink: http://dlmf.nist.gov/18.33.E11 Encodings: TeX, pMML, png See also: Annotations for §18.33(iii), §18.33 and Ch.18

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

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Keywords:: polynomials orthogonal on the unit circle, special cases
Notes:: See Askey (1982a) and Pastro (1985).
Permalink:: http://dlmf.nist.gov/18.33.iv
See also:: Annotations for §18.33 and Ch.18

Trivial

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See also:: Annotations for §18.33(iv), §18.33 and Ch.18

18.33.12	$\displaystyle\phi_{n}(z)$	$\displaystyle=z^{n},$
18.33.12	$\displaystyle w(z)$	$\displaystyle=1.$
ⓘ 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

Szegő–Askey

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Keywords:: Szeő–Askey polynomials, hypergeometric function, relations to other functions
Addition (effective with 1.2.0):: A sentence was added at the end of this paragraph.
See also:: Annotations for §18.33(iv), §18.33 and Ch.18

18.33.13		$\phi_{n}(z)=\sum_{\ell=0}^{n}\frac{{\left(\lambda+1\right)_{\ell}}{\left(% \lambda\right)_{n-\ell}}}{\ell!\,(n-\ell)!}\,z^{\ell}=\frac{{\left(\lambda% \right)_{n}}}{n!}{{}_{2}F_{1}}\left({-n,\lambda+1\atop-\lambda-n+1};z\right),$
ⓘ Symbols: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $z$ : complex variable, $\ell$ : nonnegative integer, $n$ : nonnegative integer and $\phi_{n}(z)$ : polynomials Referenced by: §18.33(v) Permalink: http://dlmf.nist.gov/18.33.E13 Encodings: TeX, pMML, png See also: Annotations for §18.33(iv), §18.33(iv), §18.33 and Ch.18

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.

Askey

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Keywords:: Askey polynomials, Rogers–Szegő polynomials, Szegő–Askey polynomials
See also:: Annotations for §18.33(iv), §18.33 and Ch.18

18.33.15		$\phi_{n}(z)=\sum_{\ell=0}^{n}\frac{\left(aq^{2};q^{2}\right)_{\ell}\left(a;q^{% 2}\right)_{n-\ell}}{\left(q^{2};q^{2}\right)_{\ell}\left(q^{2};q^{2}\right)_{n% -\ell}}(q^{-1}z)^{\ell}=\frac{\left(a;q^{2}\right)_{n}}{\left(q^{2};q^{2}% \right)_{n}}{{}_{2}\phi_{1}}\left({aq^{2},q^{-2n}\atop a^{-1}q^{2-2n}};q^{2},% \frac{qz}{a}\right),$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $z$ : complex variable, $q$ : real variable, $\ell$ : nonnegative integer, $n$ : nonnegative integer and $\phi_{n}(z)$ : polynomials Referenced by: §18.33(v) Permalink: http://dlmf.nist.gov/18.33.E15 Encodings: TeX, pMML, png See also: Annotations for §18.33(iv), §18.33(iv), §18.33 and Ch.18

with

18.33.16		$w(z)={\left\|\left(qz;q^{2}\right)_{\infty}\Bigm/\left(aqz;q^{2}\right)_{\infty% }\right\|}^{2},$
$a^{2}q^{2}<1$ .
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $w(x)$ : weight function, $z$ : complex variable and $q$ : real variable Referenced by: §18.33(v) Permalink: http://dlmf.nist.gov/18.33.E16 Encodings: TeX, pMML, png See also: Annotations for §18.33(iv), §18.33(iv), §18.33 and Ch.18

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

ⓘ

Keywords:: biorthogonal, polynomials orthogonal on the unit circle
Permalink:: http://dlmf.nist.gov/18.33.v
See also:: Annotations for §18.33 and Ch.18

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

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Keywords:: alternative set-up, polynomials orthogonal on the unit circle
Referenced by:: §18.33(i), §18.33(ii), Erratum (V1.2.0) §18.33, Version 1.2.0 (March 27, 2024)
Permalink:: http://dlmf.nist.gov/18.33.vi
Addition (effective with 1.2.0):: This subsection was added.
See also:: Annotations for §18.33 and Ch.18

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		$\int_{\|z\|=1}\Phi_{n}(z)\overline{\Phi_{m}(z)}\,\mathrm{d}\mu(z)=\kappa_{n}^{-2% }\delta_{n,m},$
ⓘ Symbols: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\overline{\NVar{z}}$ : complex conjugate, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $z$ : complex variable, $m$ : nonnegative integer and $n$ : nonnegative integer Source: Simon (2005a, (1.1.4)) Referenced by: §18.33(vi), §18.33(vi), §18.33(vi), Erratum (V1.2.0) §18.33 Permalink: http://dlmf.nist.gov/18.33.E17 Encodings: TeX, pMML, png See also: Annotations for §18.33(vi), §18.33 and Ch.18

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		$\,\mathrm{d}\mu(z)=\frac{1}{2\pi\mathrm{i}}\,w(z)\frac{\,\mathrm{d}z}{z}$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $w(x)$ : weight function and $z$ : complex variable Referenced by: §18.33(vi) Permalink: http://dlmf.nist.gov/18.33.E19 Encodings: TeX, pMML, png See also: Annotations for §18.33(vi), §18.33 and Ch.18

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

18.33.20		$\frac{1}{2\pi\mathrm{i}}\int_{\|z\|=1}\Phi_{n}(z)\overline{\Phi_{m}(z)}w(z)\frac% {\,\mathrm{d}z}{z}=\kappa_{n}^{-2}\delta_{n,m}.$
ⓘ Symbols: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\pi$ : the ratio of the circumference of a circle to its diameter, $\overline{\NVar{z}}$ : complex conjugate, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $w(x)$ : weight function, $z$ : complex variable, $m$ : nonnegative integer and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/18.33.E20 Encodings: TeX, pMML, png See also: Annotations for §18.33(vi), §18.33 and Ch.18

Recurrence Relations

ⓘ

Keywords:: Schur parameters, Szegő recurrence relations, Verblunsky coefficients, polynomials orthogonal on the unit circle, recurrence relations
Addition (effective with 1.2.0):: This paragraph was added.
See also:: Annotations for §18.33(vi), §18.33 and Ch.18

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	$\displaystyle\alpha_{n}$	$\displaystyle=-\overline{\Phi_{n+1}(0)},$
18.33.25	$\displaystyle\|\alpha_{n}\|$	$\displaystyle<1,$
ⓘ 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

ⓘ

Keywords:: Verblunsky’s theorem
Addition (effective with 1.2.0):: This paragraph was added.
See also:: Annotations for §18.33(vi), §18.33 and Ch.18

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

ⓘ

Keywords:: Szegő’s theorem
Addition (effective with 1.2.0):: This paragraph was added.
See also:: Annotations for §18.33(vi), §18.33 and Ch.18

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		$\prod_{j=0}^{\infty}\left(1-\left\|\alpha_{j}\right\|^{2}\right)=\exp\left(\frac% {1}{2\pi\mathrm{i}}\int_{\|z\|=1}\ln\left(w(z)\right)\frac{\,\mathrm{d}z}{z}% \right).$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $w(x)$ : weight function and $z$ : complex variable Proved: Simon (2005a, (1.1.18))(proved) Referenced by: §18.33(vi) Permalink: http://dlmf.nist.gov/18.33.E31 Encodings: TeX, pMML, png See also: Annotations for §18.33(vi), §18.33(vi), §18.33 and Ch.18

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		$\sum_{j=0}^{\infty}\|\alpha_{j}\|^{2}<\infty\quad\Longleftrightarrow\quad\frac{1% }{2\pi\mathrm{i}}\int_{\|z\|=1}\ln\left((w(z)\right)\frac{\,\mathrm{d}z}{z}>-\infty.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $w(x)$ : weight function and $z$ : complex variable Source: Simon (2005a, (1.1.19)) Referenced by: Erratum (V1.2.0) §18.33 Permalink: http://dlmf.nist.gov/18.33.E32 Encodings: TeX, pMML, png See also: Annotations for §18.33(vi), §18.33(vi), §18.33 and Ch.18