§18.33 Polynomials Orthogonal on the Unit Circle

Keywords:: polynomials orthogonal on the unit circle
Permalink:: http://dlmf.nist.gov/18.33

§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(i) Definition

Notes:: See Szegö (1975, (11.1.8)).
Keywords:: polynomials orthogonal on the unit circle
Permalink:: http://dlmf.nist.gov/18.33.i

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 $\frac{1}{2\pi i}\int _{{|z|=1}}\phi _{n}(z)\conj{\phi _{m}(z)}w(z)\frac{dz}{z}=\delta _{{n,m}},$

Symbols:: $\delta _{{j,k}}$ : Kronecker delta, $\conj{z}$ : complex conjugate, $dx$ : differential of $x$ , $\int$ : integral, $w(x)$ : weight function, $z$ : complex variable, $m$ : nonnegative integer, $n$ : nonnegative integer and $\phi _{n}(z)$ : polynomials
Permalink:: http://dlmf.nist.gov/18.33.E1
Encodings:: TeX, pMML, png

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

§18.33(ii) Recurrence Relations

Notes:: See Szegö (1975, (11.4.6), (11.4.7)). (18.33.6) follows from (18.33.4), (18.33.5).
Keywords:: polynomials orthogonal on the unit circle
Permalink:: http://dlmf.nist.gov/18.33.ii

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

where $\kappa _{n}(>0)$ , and $\kappa _{{n,n-\ell}}(\in\Complex)$ are constants. Also denote

18.33.3 $\phi _{n}^{{*}}(z)=\kappa _{n}z^{n}+\sum _{{\ell=1}}^{n}\conj{\kappa}_{{n,n-\ell}}z^{{n-\ell}},$

Symbols:: $\conj{z}$ : complex conjugate, $z$ : complex variable, $\ell$ : nonnegative integer, $n$ : nonnegative integer and $\phi _{n}(z)$ : polynomials
Permalink:: http://dlmf.nist.gov/18.33.E3
Encodings:: TeX, pMML, png

where the bar again signifies compex conjugate. Then

18.33.4 $\kappa _{n}z\phi _{n}(z)=\kappa _{{n+1}}\phi _{{n+1}}(z)-\phi _{{n+1}}(0)\phi _{{n+1}}^{{*}}(z),$

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

18.33.5 $\kappa _{n}\phi _{{n+1}}(z)=\kappa _{{n+1}}z\phi _{n}(z)+\phi _{{n+1}}(0)\phi _{n}^{{*}}(z),$

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

18.33.6 $\kappa _{n}\phi _{n}(0)\phi _{{n+1}}(z)+\kappa _{{n-1}}\phi _{{n+1}}(0)z\phi _{{n-1}}(z)=\left(\kappa _{n}\phi _{{n+1}}(0)+\kappa _{{n+1}}\phi _{n}(0)z\right)\phi _{n}(z).$

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

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

Notes:: See Szegö (1975, (11.5.2)). (18.33.10) and (18.33.11) follow from (18.33.8), (18.33.9).
Keywords:: polynomials orthogonal on the unit circle
Permalink:: http://dlmf.nist.gov/18.33.iii

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

18.33.7

$w_{1}(x)=(1-x^{2})^{{-\frac{1}{2}}}w\left(x+i(1-x^{2})^{{\frac{1}{2}}}\right),$

$w_{2}(x)=(1-x^{2})^{{\frac{1}{2}}}w\left(x+i(1-x^{2})^{{\frac{1}{2}}}\right).$

Symbols:: $w(x)$ : weight function, $w_{x}$ : weights and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.33.E7
Encodings:: TeX, TeX, pMML, pMML, png, png

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 $p_{n}\left(\tfrac{1}{2}(z+z^{{-1}})\right)=\text{(const.)}\times{\left(z^{{-n}}\phi _{{2n}}(z)+z^{n}\phi _{{2n}}(z^{{-1}})\right)}=\text{(const.)}\times{\left(z^{{-n+1}}\phi _{{2n-1}}(z)+z^{{n-1}}\phi _{{2n-1}}(z^{{-1}})\right)},$

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

18.33.9 $q_{n}\left(\tfrac{1}{2}(z+z^{{-1}})\right)={\text{(const.)}\times\frac{z^{{-n-1}}\phi _{{2n+2}}(z)-z^{{n+1}}\phi _{{2n+2}}(z^{{-1}})}{z-z^{{-1}}}}={\text{(const.)}\times\frac{z^{{-n}}\phi _{{2n+1}}(z)-z^{n}\phi _{{2n+1}}(z^{{-1}})}{z-z^{{-1}}}}.$

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

Conversely,

18.33.10 $z^{{-n}}\phi _{{2n}}(z)={A_{n}p_{n}\left(\tfrac{1}{2}(z+z^{{-1}})\right)+B_{n}(z-z^{{-1}})q_{{n-1}}\left(\tfrac{1}{2}(z+z^{{-1}})\right)},$

Symbols:: $z$ : complex variable, $n$ : nonnegative integer, $\phi _{n}(z)$ : polynomials, $q_{n}(x)$ : an orthogonal polynomial, $p_{n}(x)$ : an orthogonal polynomial, $A_{n}$ : coefficient and $B_{n}$ : coefficient
Referenced by:: §18.33(iii)
Permalink:: http://dlmf.nist.gov/18.33.E10
Encodings:: TeX, pMML, png

18.33.11 $z^{{-n+1}}\phi _{{2n-1}}(z)={C_{n}p_{n}\left(\tfrac{1}{2}(z+z^{{-1}})\right)+D_{n}(z-z^{{-1}})q_{{n-1}}\left(\tfrac{1}{2}(z+z^{{-1}})\right)},$

Symbols:: $z$ : complex variable, $n$ : nonnegative integer, $\phi _{n}(z)$ : polynomials, $q_{n}(x)$ : an orthogonal polynomial, $p_{n}(x)$ : an orthogonal polynomial, $C_{n}$ : coefficient and $D_{n}$ : coefficient
Referenced by:: §18.33(iii)
Permalink:: http://dlmf.nist.gov/18.33.E11
Encodings:: TeX, pMML, png

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

§18.33(iv) Special Cases

Notes:: See Askey (1982) and Pastro (1985).
Keywords:: polynomials orthogonal on the unit circle
Permalink:: http://dlmf.nist.gov/18.33.iv

¶ Trivial

18.33.12

$\phi _{n}(z)=z^{n},$

$w(z)=1.$

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

¶ Szegö–Askey

Keywords:: Szegö–Askey polynomials, hypergeometric function

18.33.13 $\phi _{n}(z)=\sum _{{\ell=0}}^{n}\frac{\left(\lambda+1\right)_{{\ell}}\left(\lambda\right)_{{n-\ell}}}{\ell!\,(n-\ell)!}\, z^{\ell}=\frac{\left(\lambda\right)_{{n}}}{n!}\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left({-n,\lambda+1\atop-\lambda-n+1};z\right),$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial, $z$ : 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

with

18.33.14

$w(z)=\left(1-\tfrac{1}{2}(z+z^{{-1}})\right)^{\lambda},$

$w_{1}(x)=(1-x)^{{\lambda-\frac{1}{2}}}(1+x)^{{-\frac{1}{2}}},$

$w_{2}(x)=(1-x)^{{\lambda+\frac{1}{2}}}(1+x)^{{\frac{1}{2}}},$ $\lambda>-\tfrac{1}{2}$ .

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

For the hypergeometric function $\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits$ see §§15.1 and 15.2(i).

¶ Askey

Keywords:: Askey polynomials, Rogers–Szegö polynomials, Szegö–Askey polynomials

18.33.15 $\phi _{n}(z)=\sum _{{\ell=0}}^{n}\frac{\left(aq^{2};q^{2}\right)_{{\ell}}\left(a;q^{2}\right)_{{n-\ell}}}{\left(q^{2};q^{2}\right)_{{\ell}}\left(q^{2};q^{2}\right)_{{n-\ell}}}(q^{{-1}}z)^{\ell}=\frac{\left(a;q^{2}\right)_{{n}}}{\left(q^{2};q^{2}\right)_{{n}}}\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left({aq^{2},q^{{-2n}}\atop a^{{-1}}q^{{2-2n}}};q^{2},\frac{qz}{a}\right),$

Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $\mathop{{{}_{{r+1}}\phi _{{s}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q,z\right)$ : 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

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:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (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

For the notation, including the basic hypergeometric function $\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits$ , 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

Keywords:: polynomials orthogonal on the unit circle
Permalink:: http://dlmf.nist.gov/18.33.v

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.