Van Vleck theorem

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21: Bibliography M

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The supports of measures associated with orthogonal polynomials and the spectra of the related selfadjoint operators. Rocky Mountain J. Math. 21 (1), pp. 501–527.

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External Links:

ISSN 0035-7596, Document, Link, MathReview (T. S. Chihara)

Referenced by:

§18.2(xi).

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Mathieusche Funktionen und Sphäroidfunktionen mit Anwendungen auf physikalische und technische Probleme. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band LXXI, Springer-Verlag, Berlin (German).

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External Links:

MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

Table 28.1.1, §28.10(i), §28.11, §28.12(i), §28.12(ii), §28.12(iii), §28.14, §28.15(i), §28.15(ii), §28.16, §28.17, §28.19, §28.2(ii), §28.2(iii), §28.2(iv), §28.2(v), §28.2(vi), §28.20(ii), §28.20(iii), §28.20(vi), §28.20(vii), §28.22(i), §28.22(ii), §28.23, §28.24, §28.25, §28.28(i), 2nd item, 3rd item, §28.33(ii), item (e), item (d), §28.4(i), §28.4(ii), §28.4(iii), §28.4(iv), §28.4(v), §28.4(vi), §28.4(vi), §28.5(i), §28.5(i), §28.6(i), §28.6(ii), §28.7, §28.7, §28.8(i), §28.8(ii), §28.9, Ch.28, §30.1, §30.10, §30.11(iii), §30.11(iii), §30.11(iv), §30.11(v), §30.11(vi), §30.11(vi), §30.13(i), §30.13(ii), §30.13(iii), §30.13(iv), §30.13(v), §30.13(v), §30.14(i), §30.14(ii), §30.14(iii), §30.14(iv), §30.14(v), §30.14(v), §30.16(i), §30.16(i), §30.2(i), §30.2(ii), §30.3(i), §30.3(ii), §30.3(iii), §30.3(iv), §30.4(i), §30.4(ii), §30.4(iv), §30.5, §30.5, §30.6, §30.6, §30.8(i), §30.8(ii), §30.9(i), §30.9(i), §30.9(ii), §30.9(ii), Ch.30, Notations P, Notations P, Notations Q, Notations Q.

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Handbook of Applied Cryptography. CRC Press, Boca Raton, FL.

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Notes:

With a foreword by Ronald L. Rivest

External Links:

ISBN 0-8493-8523-7, MathReview (Yvo Desmedt), ZentralBlatt

Referenced by:

§27.16.

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A $q$-analog of the Gauss summation theorem for hypergeometric series in $U(n)$ . Adv. in Math. 72 (1), pp. 59–131.

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External Links:

ISSN 0001-8708, Document, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§17.15.

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Jacobian Elliptic Function Tables. Dover Publications Inc., New York.

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External Links:

MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§22.20(vii).

Encodings:

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22: 18.33 Polynomials Orthogonal on the Unit Circle

… ►See Gasper (1981) and Hendriksen and van Rossum (1986) for relations with Laurent polynomials orthogonal on the unit circle. … ►

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

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Symbols:

$\overline{z}$: complex conjugate, $z$: complex variable and $n$: nonnegative integer

Proved:

Simon (2005a, Theorem 1.5.2)(proved)

Referenced by:

§18.33(vi), §18.33(vi), §18.33(vi), §18.33(vi)

Permalink:

http://dlmf.nist.gov/18.33.E23

Encodings:

pMML, png, TeX

See also:

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

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Verblunsky’s Theorem

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Szegő’s Theorem

►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 …

23: 5.10 Continued Fractions

… ►Also see Cuyt et al. (2008, pp. 223–228), Jones and Thron (1980, pp. 348–350), Lorentzen and Waadeland (1992, pp. 221–224), and Jones and Van Assche (1998).

24: 21.10 Methods of Computation

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Deconinck and van Hoeij (2001). Here a plane algebraic curve representation of the Riemann surface is used.

25: 28.36 Software

… ►See also Clemm (1969), Delft Numerical Analysis Group (1973), Rengarajan and Lewis (1980), Van Buren and Boisvert (2007), and Ziener et al. (2012).

26: Bibliography C

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The Staudt-Clausen theorem. Math. Mag. 34, pp. 131–146.

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External Links:

MathReview (K. Goldberg), ZentralBlatt

Referenced by:

§24.6.

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Short proofs of three theorems on elliptic integrals. SIAM J. Math. Anal. 9 (3), pp. 524–528.

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External Links:

Document, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§19.26(i).

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The universal von Staudt theorems. Trans. Amer. Math. Soc. 315 (2), pp. 591–603.

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External Links:

ISSN 0002-9947, FullText, MathReview (T. Metsänkylä), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

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Painlevé Equations—Nonlinear Special Functions: Computation and Application. In Orthogonal Polynomials and Special Functions, F. Marcellàn and W. van Assche (Eds.), Lecture Notes in Math., Vol. 1883, pp. 331–411.

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External Links:

ISBN 3-540-31062-2, MathReview (Marta Mazzocco), ZentralBlatt

Referenced by:

Ch.32.

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Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

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External Links:

ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§13.29(v), §15.19(v).

Encodings:

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27: Bibliography S

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Orthogonal polynomials arising in the numerical evaluation of inverse Laplace transforms. Math. Tables Aids Comput. 9 (52), pp. 164–177.

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External Links:

FullText, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§3.5(viii).

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Ein Verfahren zur Berechnung des charakteristischen Exponenten der Mathieuschen Differentialgleichung III. Numer. Math. 8 (1), pp. 68–71.

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Notes:

Includes Algol program

External Links:

ISSN 0029-599X, Document, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§28.36(ii).

Encodings:

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Ein Verfahren zur Berechnung des charakteristischen Exponenten der Mathieuschen Differentialgleichung I. Numer. Math. 3 (1), pp. 30–38.

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External Links:

ISSN 0029-599X, Document, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

item (b).

Encodings:

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Characterization of Jacobian varieties in terms of soliton equations. Invent. Math. 83 (2), pp. 333–382.

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External Links:

ISSN 0020-9910, Document, MathReview (Bert van Geemen), ZentralBlatt

Referenced by:

§21.9.

Encodings:

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Generalized Hypergeometric Functions. Cambridge University Press, Cambridge.

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External Links:

ISBN 9780521090612, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

Ch.15, §16.2(iii), §16.4(ii), §16.4(iii), §16.4(v), Ch.16, §17.1, §17.4(i), Ch.17.

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28: 34.5 Basic Properties: $\mathit{6}j$ Symbol

… ►For generating functions for the $\mathit{6}j$ symbol see Biedenharn and van Dam (1965, p. 255, eq. (4.18)). … ►They constitute addition theorems for the $\mathit{6}j$ symbol. …

29: 7.22 Methods of Computation

… ►For a comprehensive survey of computational methods for the functions treated in this chapter, see van der Laan and Temme (1984, Ch. V).

30: 27.18 Methods of Computation: Primes

… ►A practical version is described in Bosma and van der Hulst (1990). …