# orthogonal polynomials on the unit circle

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##### 1: 18.33 Polynomials Orthogonal on the Unit Circle
###### §18.33(v) Biorthogonal Polynomials on the UnitCircle
See Al-Salam and Ismail (1994) for special biorthogonal rational functions on the unit circle.
##### 2: 18.1 Notation
$\left(z_{1},\dots,z_{k};q\right)_{\infty}=\left(z_{1};q\right)_{\infty}\cdots% \left(z_{k};q\right)_{\infty}.$
##### 3: Bibliography S
• B. Simon (2005a) Orthogonal Polynomials on the Unit Circle. Part 1: Classical Theory. American Mathematical Society Colloquium Publications, Vol. 54, American Mathematical Society, Providence, RI.
• B. Simon (2005b) Orthogonal Polynomials on the Unit Circle. Part 2: Spectral Theory. American Mathematical Society Colloquium Publications, Vol. 54, American Mathematical Society, Providence, RI.
• ##### 4: 18.34 Bessel Polynomials
###### §18.34(ii) Orthogonality
Because the coefficients $C_{n}$ in (18.34.4) are not all positive, the polynomials $y_{n}\left(x;a\right)$ cannot be orthogonal on the line with respect to a positive weight function. There is orthogonality on the unit circle, however: …
##### 5: 18.35 Pollaczek Polynomials
###### §18.35(iii) Other Properties
For the polynomials $C^{(\lambda)}_{n}\left(x\right)$ and $P^{(\lambda)}_{n}\left(x;\phi\right)$ see §§18.3 and 18.19, respectively. …
##### 6: 10.59 Integrals
10.59.1 $\int_{-\infty}^{\infty}e^{ibt}\mathsf{j}_{n}\left(t\right)\mathrm{d}t=\begin{% cases}\pi i^{n}P_{n}\left(b\right),&-11,\end{cases}$
where $P_{n}$ is the Legendre polynomial18.3). …
##### 7: Bibliography
• W. A. Al-Salam and L. Carlitz (1965) Some orthogonal $q$-polynomials. Math. Nachr. 30, pp. 47–61.
• W. A. Al-Salam and M. E. H. Ismail (1994) A $q$-beta integral on the unit circle and some biorthogonal rational functions. Proc. Amer. Math. Soc. 121 (2), pp. 553–561.
• R. Askey and M. E. H. Ismail (1984) Recurrence relations, continued fractions, and orthogonal polynomials. Mem. Amer. Math. Soc. 49 (300), pp. iv+108.
• R. Askey and J. Wilson (1985) Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc. 54 (319), pp. iv+55.
• R. Askey (1975b) Orthogonal Polynomials and Special Functions. CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 21, Society for Industrial and Applied Mathematics, Philadelphia, PA.
• ##### 8: 18.10 Integral Representations
###### Jacobi
Tables 18.19.1 and 18.19.2 provide definitions via orthogonality and normalization (§§18.2(i), 18.2(iii)) for the Hahn polynomials $Q_{n}\left(x;\alpha,\beta,N\right)$, Krawtchouk polynomials $K_{n}\left(x;p,N\right)$, Meixner polynomials $M_{n}\left(x;\beta,c\right)$, and Charlier polynomials $C_{n}\left(x;a\right)$. … These polynomials are orthogonal on $(-\infty,\infty)$, and with $\Re a>0$, $\Re b>0$ are defined as follows. … These polynomials are orthogonal on $(-\infty,\infty)$, and are defined as follows. …
For further extensions, applications, and computation of orthogonal polynomials and Gauss-type formulas, see Gautschi (1994, 1996, 2004). … For the classical orthogonal polynomials related to the following Gauss rules, see §18.3. … Below we give for the classical orthogonal polynomials the recurrence coefficients $\alpha_{n}$ and $\beta_{n}$ in (3.5.30). … Complex orthogonal polynomials $p_{n}(1/\zeta)$ of degree $n=0,1,2,\dots$, in $1/\zeta$ that satisfy the orthogonality condition … …