18 Orthogonal PolynomialsOther Orthogonal Polynomials18.32 OP’s with Respect to Freud Weights18.34 Bessel Polynomials

- §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

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

18.33.1 | $$\frac{1}{2\pi \mathrm{i}}{\int}_{|z|=1}{\varphi}_{n}(z)\overline{{\varphi}_{m}(z)}w(z)\frac{dz}{z}={\delta}_{n,m},$$ | ||

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

Denote

18.33.2 | $${\varphi}_{n}(z)={\kappa}_{n}{z}^{n}+\sum _{\mathrm{\ell}=1}^{n}{\kappa}_{n,n-\mathrm{\ell}}{z}^{n-\mathrm{\ell}},$$ | ||

where ${\kappa}_{n}\phantom{\rule{0.3888888888888889em}{0ex}}(>0)$, and ${\kappa}_{n,n-\mathrm{\ell}}\phantom{\rule{0.3888888888888889em}{0ex}}(\in \u2102)$ are constants. Also denote

18.33.3 | $${\varphi}_{n}^{\ast}(z)={z}^{n}\overline{{\varphi}_{n}({\overline{z}}^{-1})}={\kappa}_{n}+\sum _{\mathrm{\ell}=1}^{n}{\overline{\kappa}}_{n,n-\mathrm{\ell}}{z}^{\mathrm{\ell}},$$ | ||

where the bar again signifies complex conjugate. Then

18.33.4 | ${\kappa}_{n}z{\varphi}_{n}(z)$ | $={\kappa}_{n+1}{\varphi}_{n+1}(z)-{\varphi}_{n+1}(0){\varphi}_{n+1}^{\ast}(z),$ | ||

18.33.5 | ${\kappa}_{n}{\varphi}_{n+1}(z)$ | $={\kappa}_{n+1}z{\varphi}_{n}(z)+{\varphi}_{n+1}(0){\varphi}_{n}^{\ast}(z),$ | ||

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

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

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

18.33.7 | ${w}_{1}(x)$ | $={(1-{x}^{2})}^{-\frac{1}{2}}w\left(x+\mathrm{i}{(1-{x}^{2})}^{\frac{1}{2}}\right),$ | ||

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

Let $\{{p}_{n}(x)\}$ and $\{{q}_{n}(x)\}$, $n=0,1,\mathrm{\dots}$, be OP’s with weight functions ${w}_{1}(x)$ and ${w}_{2}(x)$, respectively, on $(-1,1)$. Then

18.33.8 | ${p}_{n}\left(\frac{1}{2}(z+{z}^{-1})\right)$ | $\begin{array}{ll}& =\text{(const.)}\times \left({z}^{-n}{\varphi}_{2n}(z)+{z}^{n}{\varphi}_{2n}({z}^{-1})\right)\\ & =\text{(const.)}\times \left({z}^{-n+1}{\varphi}_{2n-1}(z)+{z}^{n-1}{\varphi}_{2n-1}({z}^{-1})\right),\end{array}$ | ||

18.33.9 | ${q}_{n}\left(\frac{1}{2}(z+{z}^{-1})\right)$ | $\begin{array}{ll}& =\text{(const.)}\times {\displaystyle \frac{{z}^{-n-1}{\varphi}_{2n+2}(z)-{z}^{n+1}{\varphi}_{2n+2}({z}^{-1})}{z-{z}^{-1}}}\\ & =\text{(const.)}\times {\displaystyle \frac{{z}^{-n}{\varphi}_{2n+1}(z)-{z}^{n}{\varphi}_{2n+1}({z}^{-1})}{z-{z}^{-1}}}.\end{array}$ | ||

Conversely,

18.33.10 | ${z}^{-n}{\varphi}_{2n}(z)$ | $={A}_{n}{p}_{n}\left(\frac{1}{2}(z+{z}^{-1})\right)+{B}_{n}(z-{z}^{-1}){q}_{n-1}\left(\frac{1}{2}(z+{z}^{-1})\right),$ | ||

18.33.11 | ${z}^{-n+1}{\varphi}_{2n-1}(z)$ | $={C}_{n}{p}_{n}\left(\frac{1}{2}(z+{z}^{-1})\right)+{D}_{n}(z-{z}^{-1}){q}_{n-1}\left(\frac{1}{2}(z+{z}^{-1})\right),$ | ||

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 ${q}_{n}\left(\frac{1}{2}(z+{z}^{-1})\right)$ in terms of ${\varphi}_{2n}({z}^{\pm 1})$ or ${\varphi}_{2n+1}({z}^{\pm 1})$. After a quadratic transformation (18.2.23) this would express OP’s on $[-1,1]$ with an even orthogonality measure in terms of the ${\varphi}_{n}$. See Zhedanov (1998, §2).

18.33.12 | ${\varphi}_{n}(z)$ | $={z}^{n},$ | ||

$w(z)$ | $=1.$ | |||

18.33.13 | $${\varphi}_{n}(z)=\sum _{\mathrm{\ell}=0}^{n}\frac{{\left(\lambda +1\right)}_{\mathrm{\ell}}{\left(\lambda \right)}_{n-\mathrm{\ell}}}{\mathrm{\ell}!(n-\mathrm{\ell})!}{z}^{\mathrm{\ell}}=\frac{{\left(\lambda \right)}_{n}}{n!}{}_{2}{}^{}F_{1}^{}(\genfrac{}{}{0pt}{}{-n,\lambda +1}{-\lambda -n+1};z),$$ | ||

with

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

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

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

$\lambda >-\frac{1}{2}$. | ||||

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 | $${\varphi}_{n}(z)=\sum _{\mathrm{\ell}=0}^{n}\frac{{(a{q}^{2};{q}^{2})}_{\mathrm{\ell}}{(a;{q}^{2})}_{n-\mathrm{\ell}}}{{({q}^{2};{q}^{2})}_{\mathrm{\ell}}{({q}^{2};{q}^{2})}_{n-\mathrm{\ell}}}{({q}^{-1}z)}^{\mathrm{\ell}}=\frac{{(a;{q}^{2})}_{n}}{{({q}^{2};{q}^{2})}_{n}}{}_{2}{}^{}\varphi _{1}^{}(\genfrac{}{}{0pt}{}{a{q}^{2},{q}^{-2n}}{{a}^{-1}{q}^{2-2n}};{q}^{2},\frac{qz}{a}),$$ | ||

with

18.33.16 | $$w(z)={\left|{(qz;{q}^{2})}_{\mathrm{\infty}}/{(aqz;{q}^{2})}_{\mathrm{\infty}}\right|}^{2},$$ | ||

$$. | |||

For the notation, including the basic hypergeometric function ${}_{2}{}^{}\varphi _{1}^{}$, 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).

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.

Instead of orthonormal polynomials $\{{\varphi}_{n}(z)\}$
Simon (2005a, b) uses *monic* polynomials
${\mathrm{\Phi}}_{n}(z)$.
Let $\mu $ be a probability measure on the unit circle of which
the support is an infinite set.
A system of monic polynomials $\{{\mathrm{\Phi}}_{n}(z)\}$, $n=0,1,\mathrm{\dots}$,
where ${\mathrm{\Phi}}_{n}(x)$ is of proper degree $n$,
is *orthogonal on the unit circle with respect
to the measure* $\mu $ if

18.33.17 | $${\int}_{|z|=1}{\mathrm{\Phi}}_{n}(z)\overline{{\mathrm{\Phi}}_{m}(z)}d\mu (z)={\kappa}_{n}^{-2}{\delta}_{n,m},$$ | ||

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

18.33.18 | $${\varphi}_{n}(z)={\kappa}_{n}{\mathrm{\Phi}}_{n}(z).$$ | ||

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

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

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

18.33.20 | $$\frac{1}{2\pi \mathrm{i}}{\int}_{|z|=1}{\mathrm{\Phi}}_{n}(z)\overline{{\mathrm{\Phi}}_{m}(z)}w(z)\frac{dz}{z}={\kappa}_{n}^{-2}{\delta}_{n,m}.$$ | ||

For a polynomial

18.33.21 | $$p(z)=\sum _{k=0}^{n}{c}_{k}{z}^{k},$$ | ||

${c}_{n}\ne 0$, | |||

with complex coefficients ${c}_{k}$ and of a certain degree $n$ define the reversed polynomial ${p}^{\ast}(z)$ by

18.33.22 | $${p}^{\ast}(z)\equiv {z}^{n}\overline{p({\overline{z}}^{-1})}=\sum _{k=0}^{n}\overline{{c}_{n-k}}{z}^{k}.$$ | ||

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

18.33.23 | ${\mathrm{\Phi}}_{n+1}(z)$ | $=z{\mathrm{\Phi}}_{n}(z)-\overline{{\alpha}_{n}}{\mathrm{\Phi}}_{n}^{\ast}(z),$ | ||

18.33.24 | ${\mathrm{\Phi}}_{n+1}^{\ast}(z)$ | $={\mathrm{\Phi}}_{n}^{\ast}(z)-{\alpha}_{n}z{\mathrm{\Phi}}_{n}(z).$ | ||

Then

18.33.25 | ${\alpha}_{n}$ | $=-\overline{{\mathrm{\Phi}}_{n+1}(0)},$ | ||

$|{\alpha}_{n}|$ | $$ | |||

18.33.26 | ${\rho}_{n}$ | $\equiv \sqrt{1-{|{\alpha}_{n}|}^{2}}={\displaystyle \frac{{\kappa}_{n}}{{\kappa}_{n+1}}}.$ | ||

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

18.33.27 | $z{\mathrm{\Phi}}_{n}(z)$ | $={\rho}_{n}^{-2}\left({\mathrm{\Phi}}_{n+1}(z)+\overline{{\alpha}_{n}}{\mathrm{\Phi}}_{n+1}^{\ast}(z)\right),$ | ||

18.33.28 | ${\mathrm{\Phi}}_{n}^{\ast}(z)$ | $={\rho}_{n}^{-2}\left({\mathrm{\Phi}}_{n+1}^{\ast}(z)+{\alpha}_{n}{\mathrm{\Phi}}_{n+1}(z)\right).$ | ||

Combination of (18.33.23) and (18.33.24) gives

18.33.29 | $${\mathrm{\Phi}}_{n+1}(z)\pm {\mathrm{\Phi}}_{n+1}^{\ast}(z)=(1\mp {\alpha}_{n})z{\mathrm{\Phi}}_{n}(z)+(\pm 1-\overline{{\alpha}_{n}}){\mathrm{\Phi}}_{n}^{\ast}(z),$$ | ||

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

18.33.30 | $$\overline{{\alpha}_{n-1}}{\mathrm{\Phi}}_{n+1}(z)=\left(\overline{{\alpha}_{n}}+\overline{{\alpha}_{n-1}}z\right){\mathrm{\Phi}}_{n}(z)-\overline{{\alpha}_{n}}{\rho}_{n-1}^{2}z{\mathrm{\Phi}}_{n-1}(z)$$ | ||

for $n>0$, while ${\mathrm{\Phi}}_{1}(z)=z-\overline{{\alpha}_{0}}$.

This states that for any sequence ${\{{\alpha}_{n}\}}_{n=0}^{\mathrm{\infty}}$ with ${\alpha}_{n}\in \u2102$ and $$ the polynomials ${\mathrm{\Phi}}_{n}(z)$ generated by the recurrence relations (18.33.23), (18.33.24) with ${\mathrm{\Phi}}_{0}(z)=1$ 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)).

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 | $$\prod _{j=0}^{\mathrm{\infty}}\left(1-{\left|{\alpha}_{j}\right|}^{2}\right)=\mathrm{exp}\left(\frac{1}{2\pi \mathrm{i}}{\int}_{|z|=1}\mathrm{ln}\left(w(z)\right)\frac{dz}{z}\right).$$ | ||

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

18.33.32 | $$ | ||