Szeg? recurrence relations

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1: 18.33 Polynomials Orthogonal on the Unit Circle

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

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Symbols:

$\overline{z}$: complex conjugate, $\equiv$: equals by definition, $z$: complex variable, $k$: nonnegative integer and $n$: nonnegative integer

Source:

Simon (2005a, (1.1.7))

Permalink:

http://dlmf.nist.gov/18.33.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §18.33(vi), §18.33(vi), §18.33 and Ch.18

►The Verblunsky coefficients (also called Schur parameters or reflection coefficients) are the coefficients ${\alpha }_n$ in the Szegő recurrence relations … ►Equivalent to the recurrence relations (18.33.23), (18.33.24) are the inverse Szegő recurrence relations …

2: 18.9 Recurrence Relations and Derivatives

§18.9 Recurrence Relations and Derivatives

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§18.9(i) Recurrence Relations

… ►with initial values $p_0(x)=1$ and $p_1(x)=A_{0}x+B_0$. … ►with initial values $p_0(x)=1$ and $p_1(x)=a_0^{-1}(x-b_0)$. … ►and the structure relation …

3: 18.2 General Orthogonal Polynomials

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§18.2(iv) Recurrence Relations

… ►the monic recurrence relations (18.2.8) and (18.2.10) take the form … ►Then, with the coefficients (18.2.11_4) associated with the monic OP’s $p_n$, the orthonormal recurrence relation for $q_n$ takes the form … ►The recurrence relations (18.2.10) can be equivalently written as … ►More generally, §18.30 defines the recurrence relation of the $c$th associated monic OP by means of a similar shift by $c$ in (18.2.11_5). …

4: 18.35 Pollaczek Polynomials

… ►The three types of Pollaczek polynomials were successively introduced in Pollaczek (1949a, b, 1950), see also Erdélyi et al. (1953b, p.219) and, for type 1 and 2, Szegö (1950) and Askey (1982b). … ►The Pollaczek polynomials of type 3 are defined by the recurrence relation (in first form (18.2.8)) …the recurrence relation of form (18.2.11_5) becomes … ►As in the coefficients of the above recurrence relations $n$ and $c$ only occur in the form $n+c$, the type 3 Pollaczek polynomials may also be called the associated type 2 Pollaczek polynomials by using the terminology of §18.30. … ► …

5: Bibliography

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Recurrence relations for the Fresnel integral ${\int }_0^{\mathrm{\infty }}\frac{\exp (-ct)dt}{\sqrt{t}(1+t^2)}$ and similar integrals. Comm. ACM 17 (8), pp. 480–481.

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External Links:

ISSN 0001-0782, Document, MathReview, ZentralBlatt

Referenced by:

§6.18(ii).

Encodings:

BibTeX

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Szegő Type Asymptotics for the Reproducing Kernel in Spaces of Full-Plane Weighted Polynomials. Comm. Math. Phys. 398 (3), pp. 1291–1348.

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External Links:

ISSN 0010-3616, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§8.11(iii), §8.11(iii), Erratum (V1.1.10) for Section 8.11(iii).

Encodings:

BibTeX

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Integral equations and relations for Lamé functions. Quart. J. Math. Oxford Ser. (2) 15, pp. 103–115.

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External Links:

Document, MathReview (I. Marx), ZentralBlatt

Referenced by:

§29.8.

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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Commentary on the Paper “Beiträge zur Theorie der Toeplitzschen Form”. In Gábor Szegő, Collected Papers. Vol. 1, Contemporary Mathematicians, pp. 303–305.

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External Links:

ISBN 3-7643-3056-2, MathReview (G. Piranian)

Referenced by:

§18.33(iv), §18.33(iv), §18.33(v).

Encodings:

BibTeX

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6: Bibliography W

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Asymptotics of orthogonal polynomials via recurrence relations. Anal. Appl. (Singap.) 10 (2), pp. 215–235.

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External Links:

ISSN 0219-5305, Document, Link, MathReview (Li Zhong Peng), ZentralBlatt

Referenced by:

§2.9(iii), §2.9(iii).

Encodings:

BibTeX

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Some useful integrals of $\mathrm{Si}(x),$ $\mathrm{Ci}(x)$ and related integrals. Optica Acta 14 (3), pp. 317–322.

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External Links:

Document, MathReview (S. D. Bajpai)

Referenced by:

§6.14(iii), §6.7(iv).

Encodings:

BibTeX

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Hypergeometric Series, Recurrence Relations and Some New Orthogonal Polynomials. Ph.D. Thesis, University of Wisconsin, Madison, WI.

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Referenced by:

§16.4(iii), §16.4(iii), §16.4(iii).

Encodings:

BibTeX

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Computation with Recurrence Relations. Pitman, Boston, MA.

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External Links:

ISBN 0-273-08508-5, MathReview (S. Conde), ZentralBlatt

Referenced by:

§13.29(iv), §16.25, §2.9(iii), §3.10(iii), §3.6(iii), §3.6(v), §3.6(vii).

Encodings:

BibTeX

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Asymptotics of linear recurrences. Anal. Appl. (Singap.) 12 (4), pp. 463–484.

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External Links:

ISSN 0219-5305, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§2.9(iii), §2.9(iii).

Encodings:

BibTeX

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7: Bibliography C

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A recurrence formula for $\zeta (2n)$ . Proc. Amer. Math. Soc. 12 (6), pp. 991–992.

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External Links:

FullText, MathReview (R. D. James)

Referenced by:

§24.14(i).

Encodings:

BibTeX

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Some inequalities for hypergeometric functions. Proc. Amer. Math. Soc. 17 (1), pp. 32–39.

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External Links:

FullText, MathReview (Gabor Szegő), ZentralBlatt

Referenced by:

§19.24(i), §19.24(i), §19.24(ii), §19.9(i), §19.9(ii).

Encodings:

BibTeX

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Table of integrals of squared Jacobian elliptic functions and reductions of related hypergeometric $R$-functions. Math. Comp. 75 (255), pp. 1309–1318.

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External Links:

ISSN 0025-5718, Document, MathReview, ZentralBlatt

Referenced by:

§19.25(v), §19.29(i), §22.14(ii).

Encodings:

BibTeX

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A Unified Approach to Recurrence Algorithms. In Approximation and Computation (West Lafayette, IN, 1993), R. V. M. Zahar (Ed.), International Series of Computational Mathematics, Vol. 119, pp. 97–120.

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External Links:

ISBN 0-8176-3753-2, MathReview (R. M. M. Mattheij), ZentralBlatt

Referenced by:

§3.6(vii).

Encodings:

BibTeX

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Numerical integration of related Hankel transforms by quadrature and continued fraction expansion. Geophysics 48 (12), pp. 1671–1686.

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Notes:

Fortran code.

External Links:

ISSN 0098-3500, Document

Referenced by:

§10.77(ix).

Encodings:

BibTeX

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