orthogonal polynomials on the unit circle

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1: 18.33 Polynomials Orthogonal on the Unit Circle

§18.33 Polynomials Orthogonal on the Unit Circle

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§18.33(i) Definition

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

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§18.33(v) Biorthogonal Polynomials on the Unit Circle

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Recurrence Relations

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2: 18.1 Notation

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$${(z_1,\mathrm{\dots },z_k;q)}_{\mathrm{\infty }}={(z_1;q)}_{\mathrm{\infty }}\mathrm{\cdots }{(z_k;q)}_{\mathrm{\infty }}.$$

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3: Bibliography Z

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On some classes of polynomials orthogonal on arcs of the unit circle connected with symmetric orthogonal polynomials on an interval. J. Approx. Theory 94 (1), pp. 73–106.

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External Links:

ISSN 0021-9045, Link, MathReview (T. S. Chihara), ZentralBlatt

Referenced by:

§18.33(iii).

Encodings:

BibTeX

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4: Bibliography S

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Orthogonal Polynomials on the Unit Circle. Part 1: Classical Theory. American Mathematical Society Colloquium Publications, Vol. 54, American Mathematical Society, Providence, RI.

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External Links:

ISBN 0-8218-3446-0, MathReview (P. L. Duren), ZentralBlatt

Referenced by:

(18.33.17), (18.33.18), (18.33.21), (18.33.22), (18.33.23), (18.33.24), (18.33.25), (18.33.26), (18.33.27), (18.33.28), (18.33.30), (18.33.31), (18.33.32), §18.33(vi), §18.33(i), §18.33(vi).

Encodings:

BibTeX

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Orthogonal Polynomials on the Unit Circle. Part 2: Spectral Theory. American Mathematical Society Colloquium Publications, Vol. 54, American Mathematical Society, Providence, RI.

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External Links:

ISBN 0-8218-3675-7, MathReview (P. L. Duren), ZentralBlatt

Referenced by:

§18.33(i), §18.33(vi).

Encodings:

BibTeX

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5: Errata

… ►In regard to orthogonal polynomials on the unit circle, we now discuss monic polynomials, Verblunsky’s Theorem, and Szegő’s theorem. …

6: 18.34 Bessel Polynomials

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§18.34(ii) Orthogonality

… ►Hence the full system of polynomials $y_n(x;a)$ cannot be orthogonal on the line with respect to a positive weight function, but this is possible for a finite system of such polynomials, the Romanovski–Bessel polynomials, if : … ►Orthogonality of the full system on the unit circle can be given with a much simpler weight function: …See Ismail (2009, (4.10.9)) for orthogonality on the unit circle for general values of $a$. … ►In this limit the finite system of Jacobi polynomials $P_n^{(\alpha ,\beta )}\left(x\right)$ which is orthogonal on $(1,\mathrm{\infty })$ (see §18.3) tends to the finite system of Romanovski–Bessel polynomials which is orthogonal on $(0,\mathrm{\infty })$ (see (18.34.5_5)). …

7: Bibliography

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Some orthogonal $q$-polynomials. Math. Nachr. 30, pp. 47–61.

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External Links:

ISSN 0025-584X, Document, MathReview (A. E. Danese), ZentralBlatt

Referenced by:

§18.27(vii).

Encodings:

BibTeX

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Characterization theorems for orthogonal polynomials. In Orthogonal Polynomials (Columbus, OH, 1989), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 294, pp. 1–24.

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External Links:

Link, MathReview (Walter Van Assche), ZentralBlatt

Referenced by:

§18.3.

Encodings:

BibTeX

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Classical Orthogonal Polynomials. In Orthogonal Polynomials and Applications, C. Brezinski, A. Draux, A. P. Magnus, P. Maroni, and A. Ronveaux (Eds.), Lecture Notes in Math., Vol. 1171, pp. 36–62.

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External Links:

MathReview, ZentralBlatt

Referenced by:

(18.27.12_5), (18.27.6).

Encodings:

BibTeX

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Recurrence relations, continued fractions, and orthogonal polynomials. Mem. Amer. Math. Soc. 49 (300), pp. iv+108.

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External Links:

ISSN 0065-9266, MathReview (W. A. Al-Salam), ZentralBlatt

Referenced by:

§18.28(iv).

Encodings:

BibTeX

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Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc. 54 (319), pp. iv+55.

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External Links:

ISSN 0065-9266, MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§18.21(ii), §18.22(ii), (18.28.24), §18.28(ii), Ch.18, Profile Richard A. Askey.

Encodings:

BibTeX

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8: 10.59 Integrals

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10.59.1

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind and $n$: integer

A&S Ref:

11.4.26

Referenced by:

§10.59

Permalink:

http://dlmf.nist.gov/10.59.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.59 and Ch.10

►where $P_n$ is the Legendre polynomial (§18.3). …

9: 37.6 Plane with Weight Function ${\mathrm{e}}^{-x^2-y^2}$

… ►The spaces ${𝒱}_n$ are eigenspaces of a second order partial differential operator, see (37.6.12). … ►Then the polynomials $S_{m,n-m}(x+\mathrm{i}y,x-\mathrm{i}y)$ ($m=0,1,\mathrm{\dots },n$) form an orthogonal basis of the space ${𝒱}_n\otimes ℂ$ of complex-valued orthogonal polynomials of degree $n$ on ${ℝ}^2$ with weight function ${\mathrm{e}}^{-x^2-y^2}$. … ►In particular, for the orthogonal basis (37.6.2) of products of Hermite polynomials, …and, for the orthogonal basis (37.6.3) of complex circular Hermite polynomials, … ►The explicit basis functions in §37.4 of (bi)orthogonal polynomials on the unit disk for the weight function (37.4.2) all tend after rescaling, as $\alpha \to \mathrm{\infty }$, to basis functions given above of OPs on ${ℝ}^2$ for the weight function ${\mathrm{e}}^{-x^2-y^2}$: …

10: 18.35 Pollaczek Polynomials

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18.35.7 $${(1-z{\mathrm{e}}^{\mathrm{i}\theta })}^{-\lambda +\mathrm{i}{\tau }_{a,b}(\theta )}{(1-z{\mathrm{e}}^{-\mathrm{i}\theta })}^{-\lambda -\mathrm{i}{\tau }_{a,b}(\theta )}=\sum _{n=0}^{\mathrm{\infty }}{P}_n^{(\lambda )}(\cos \theta ;a,b)z^n,$$ , .

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Symbols:

$P_n^{(\lambda )}(x;a,b)$: Pollaczek polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $n$: nonnegative integer and ${\tau }_{a,b}(\theta )$

Proved:

Ismail (2009, (5.4.8))(proved)

Permalink:

http://dlmf.nist.gov/18.35.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §18.35(iii), §18.35 and Ch.18

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