# §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. See Simon (2005a, b) for general theory.

## §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 compex 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

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

Assume that $w(e^{\mathrm{i}\phi})=w(e^{-\mathrm{i}\phi})$. 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, $\phi_{n}(z)$: polynomials and $p_{n}(x)$: an orthogonal polynomial 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, $\phi_{n}(z)$: polynomials and $q_{n}(x)$: an orthogonal polynomial Referenced by: §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 $z$.

## §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).

 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.

## §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.