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1: 25.10 Zeros

… ►More than 41% of all the zeros in the critical strip lie on the critical line (Bui et al. (2011)). …

2: Bibliography B

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Applied Combinatorial Mathematics. Robert E. Krieger, Malabar, FL.

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External Links:

ZentralBlatt

Referenced by:

§26.19, §26.20.

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Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model. Ann. of Math. (2) 150 (1), pp. 185–266.

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External Links:

ISSN 0003-486X, Document, MathReview (Thomas Kriecherbauer), ZentralBlatt

Referenced by:

§18.32.

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Occurrence of periodic Lamé functions at bifurcations in chaotic Hamiltonian systems. J. Phys. A 34 (40), pp. 8199–8220.

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External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§29.19(i).

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More than 41% of the zeros of the zeta function are on the critical line. Acta Arith. 150 (1), pp. 35–64.

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ISSN 0065-1036, Document, Link, MathReview (Timothy S. Trudgian), ZentralBlatt

Referenced by:

§25.10(ii), §25.10(ii), Erratum (V1.1.4) for Subsection 25.10(ii).

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Euler functions $E_{\alpha }(z)$ with complex $\alpha$ and applications. In Approximation, probability, and related fields (Santa Barbara, CA, 1993), G. Anastassiou and S. T. Rachev (Eds.), pp. 127–150.

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External Links:

MathReview (R. N. Kalia), ZentralBlatt

Referenced by:

§24.16(iii).

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3: Errata

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In the paragraph immediately below (25.10.4), it was originally stated that “more than one-third of all zeros in the critical strip lie on the critical line.” which referred to Levinson (1974). This sentence has been updated with “one-third” being replaced with “41%” now referring to Bui et al. (2011) (suggested by Gergő Nemes on 2021-08-23).

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4: 30.7 Graphics

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Figure 30.7.3: Eigenvalues ${\lambda }_n^5\left({\gamma }^2\right)$, $n=5,6,7,8$, $-40\le {\gamma }^2\le 40$. Magnify

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Figure 30.7.4: Eigenvalues ${\lambda }_n^{10}\left({\gamma }^2\right)$, $n=10,11,12,13$, $-50\le {\gamma }^2\le 150$. Magnify

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5: Bibliography

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Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics. Comput. Phys. Comm. 150 (1), pp. 1–20.

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Notes:

Includes algorithms for Lerch’s transcendent. C and Mathematica codes by Aksenov and Jentschura are available.

External Links:

Code, Document

Referenced by:

1st item.

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The $J$-Matrix Method. Developments and Applications. Springer-Verlag.

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External Links:

ISBN 978-1-4020-6072-4

Referenced by:

§18.39(iv).

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Frequency response characteristics of the multiport planar elliptic patch. IEEE Trans. Microwave Theory Tech. 40 (8), pp. 1726–1730.

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External Links:

Document

Referenced by:

6th item.

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Sturm-Liouville Theory. Birkhäuser Verlag, Basel.

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Notes:

Past and present, Including papers from the International Colloquium held at the University of Geneva, Geneva, September 15–19, 2003

External Links:

ISBN 978-3-7643-7066-4; 3-7643-7066-1, Document, Link, MathReview (Anton Zettl)

Referenced by:

§1.13(viii), §1.18(x).

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Special Functions: Group Theoretical Aspects and Applications. D. Reidel Publishing Co., Dordrecht.

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Notes:

Reprinted by Kluwer Academic Publishers, Boston, 2001.

External Links:

ISBN 90-277-1822-9; 1-4020-0319-6, MathReview, ZentralBlatt

Referenced by:

Profile Richard A. Askey, Profile Tom H. Koornwinder.

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6: Bibliography T

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Computational aspects of incomplete gamma functions with large complex parameters. In Approximation and Computation. A Festschrift in Honor of Walter Gautschi, R. V. M. Zahar (Ed.), International Series of Numerical Mathematics, Vol. 119, pp. 551–562.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§8.11(iii), §8.20(ii), §8.25(i), §8.25(iii), 2nd item, 3rd item.

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The universal Askey-Wilson algebra and DAHA of type $(C_1^{\vee },C_1)$ . SIGMA 9, pp. Paper 047, 40 pp..

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ISSN 1815-0659, Link, MathReview (Renato Álvarez-Nodarse), ZentralBlatt

Referenced by:

§18.38(iii).

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Representation Theory and Harmonic Analysis. Contemporary Mathematics, Vol. 191, American Mathematical Society, Providence, RI.

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Notes:

Papers from the conference held in honor of Ray A. Kunze at the AMS Special Session, Cincinnati, Ohio, January 12–14, 1994

External Links:

ISBN 0-8218-0310-7, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

Profile Donald St. P. Richards.

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Sul comportamento asintotico dell’$n$-esimo polinomio di Laguerre nell’intorno dell’ascissa $4n$ . Comment. Math. Helv. 22, pp. 150–167.

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External Links:

ISSN 0010-2571, Document, MathReview (G. Szegő), ZentralBlatt

Referenced by:

§18.16(iv).

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Funzioni Ellittiche. 2nd edition, Nicola Zanichelli Editore, Bologna (Italian).

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Notes:

German edition (M. Krafft, Ed.) Akad. Verlag., Leipzig, 1948

External Links:

MathReview (Z. Nehari), ZentralBlatt

Referenced by:

Ch.19.

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7: Bibliography K

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Periodic Orbits, Spectral Statistics, and the Riemann Zeros. In Supersymmetry and Trace Formulae: Chaos and Disorder, J. P. Keating, D. E. Khmelnitskii, and I. V. Lerner (Eds.), pp. 1–15.

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Referenced by:

§25.17.

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Elliptic asymptotics of the first and second Painlevé transcendents. Uspekhi Mat. Nauk 49 (1(295)), pp. 77–140 (Russian).

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Notes:

In Russian. English translation: Russian Math. Surveys 49(1994), no. 1, pp. 81–150.

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ISSN 0042-1316, Document, MathReview (V. A. Golubeva), ZentralBlatt

Referenced by:

§32.11(i), §32.12(ii).

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On the complex zeros of $H_{\mu }(z),J_{\mu }^{\prime }(z),J_{\mu }^{\prime \prime }(z)$ for real or complex order. J. Comput. Appl. Math. 40 (3), pp. 337–344.

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ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§10.21(ix).

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Rays and caustics as physical objects. In Progress in Optics, E. Wolf (Ed.), Vol. 26, pp. 227–348.

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External Links:

ISBN 0444870962

Referenced by:

§36.14(i).

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Algebras of Virasoro type, the energy-momentum tensor, and operator expansions on Riemann surfaces. Funktsional. Anal. i Prilozhen. 23 (1), pp. 24–40 (Russian).

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Notes:

In Russian. English translation: Funct. Anal. Appl. 23(1989), no. 1, pp. 19–33

External Links:

ISSN 0374-1990, Document, MathReview (Fedor Malikov), ZentralBlatt

Referenced by:

§20.12(ii).

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8: Bibliography D

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Plancherel-Rotach asymptotic expansion for some polynomials from indeterminate moment problems. Constr. Approx. 40 (1), pp. 61–104.

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ISSN 0176-4276, Document, Link, MathReview (Francisco Marcellán), ZentralBlatt

Referenced by:

§2.9(iii), §2.9(iii).

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Pólya’s Theory of Counting. In Applied Combinatorial Mathematics, E. F. Beckenbach (Ed.), pp. 144–184.

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External Links:

ISBN 0898741726

Referenced by:

§26.20.

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Sur les zéros imaginaires des polynômes de Bernoulli. C. R. Acad. Sci. Paris Sér. I Math. 304 (6), pp. 147–150 (French).

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External Links:

ISSN 0249-6291, MathReview, ZentralBlatt

Referenced by:

§24.12(iii).

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Special Functions. World Scientific Publishing Co., Inc., River Edge, NJ.

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External Links:

ISBN 981-02-4393-6, Document, Link, MathReview Entry

Referenced by:

Profile Mourad E.H. Ismail.

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Coulomb wave functions with complex values of the variable and the parameters. J. Math. Phys. 40 (12), pp. 6145–6166.

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External Links:

ISSN 0022-2488, Document, MathReview (Raj Wilson), ZentralBlatt

Referenced by:

§33.13.

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9: 5.10 Continued Fractions

… ►For exact values of $a_7$ to $a_{11}$ and 40S values of $a_0$ to $a_{40}$, see Char (1980). …

10: 24.20 Tables

… ►In Wagstaff (2002) these results are extended to $n=60(2)152$ and $n=40(2)88$, respectively, with further complete and partial factorizations listed up to $n=300$ and $n=200$, respectively. …