self-adjoint extensions of a symmetric operator

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11: Jim Pitman

… ► 1949 in Tasmania) is a professor in the departments of statistics and mathematics at the University of California, Berkeley. …Pitman holds a B. …in statistics from the Australian National University, Canberra, and a Ph. … ►He has published extensively on probability, stochastic processes, combinatorics and is a champion for open access to resources in mathematics. ►In November 2015, Pitman was named a Senior Associate Editor of the DLMF.

12: 4.46 Tables

… ►Extensive numerical tables of all the elementary functions for real values of their arguments appear in Abramowitz and Stegun (1964, Chapter 4). … ►For 10S values of the first five complex roots of $\sin z=az$, $\cos z=az$, and $\cosh z=az$, for selected positive values of $a$, see Fettis (1976). …

13: Tom H. Koornwinder

… ►During 1968–1992 he was a Researcher in the Centrum Wiskunde & Informatica (CWI), Amsterdam. ►Koornwinder has published numerous papers on special functions, harmonic analysis, Lie groups, quantum groups, computer algebra, and their interrelations, including an interpretation of Askey–Wilson polynomials on quantum SU(2), and a five-parameter extension (the Macdonald–Koornwinder polynomials) of Macdonald’s polynomials for root systems BC. … A. … ►Koornwinder served as a Validator for the original release and publication in May 2010 of the NIST Digital Library of Mathematical Functions and the NIST Handbook of Mathematical Functions. …

14: 17.17 Physical Applications

… ►In exactly solved models in statistical mechanics (Baxter (1981, 1982)) the methods and identities of §17.12 play a substantial role. … ►Quantum groups also apply $q$-series extensively. …They were given this name because they play a role in quantum physics analogous to the role of Lie groups and special functions in classical mechanics. … ►A substantial literature on $q$-deformed quantum-mechanical Schrödinger equations has developed recently. …

15: 34.10 Zeros

… ►In a $\mathit{3}j$ symbol, if the three angular momenta $j_1,j_2,j_3$ do not satisfy the triangle conditions (34.2.1), or if the projective quantum numbers do not satisfy (34.2.3), then the $\mathit{3}j$ symbol is zero. … ►For further information, including examples of nontrivial zeros and extensions to $\mathit{9}j$ symbols, see Srinivasa Rao and Rajeswari (1993, pp. 133–215, 294–295, 299–310).

16: 8.23 Statistical Applications

… ►The functions $P(a,x)$ and $Q(a,x)$ are used extensively in statistics as the probability integrals of the gamma distribution; see Johnson et al. (1994, pp. 337–414). …The function ${\mathrm{B}}_x(a,b)$ and its normalization $I_x(a,b)$ play a similar role in statistics in connection with the beta distribution; see Johnson et al. (1995, pp. 210–275). In queueing theory the Erlang loss function is used, which can be expressed in terms of the reciprocal of $Q(a,x)$; see Jagerman (1974) and Cooper (1981, pp. 80, 316–319). …

17: Bibliography S

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An Introduction to the Operations with Series. 2nd edition, Chelsea Publishing Co., New York.

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Notes:

This is a reprint, with corrections, of the first edition [University of Pennsylvania Press, Philadelphia, Pa., 1924].

External Links:

MathReview, ZentralBlatt

Referenced by:

§24.1, §24.6.

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Operators with Singular Continuous Spectrum: I. General Operators. Annals of Mathematics 141 (1), pp. 131–145.

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External Links:

MathReview Entry

Referenced by:

§1.18(vii).

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Prolate spheroidal wave functions, Fourier analysis and uncertainity. IV. Extensions to many dimensions; generalized prolate spheroidal functions. Bell System Tech. J. 43, pp. 3009–3057.

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External Links:

ISSN 0005-8580, MathReview (J. Meixner), ZentralBlatt

Referenced by:

§30.12.

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A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.

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Notes:

For software see John Stembridge’s home page. The SF package contains Maple software for zonal, Jack, and Macdonald polynomials.

External Links:

Home Page, ISSN 0747-7171, Document, MathReview, ZentralBlatt

Referenced by:

3rd item.

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An extension of Airy’s equation. SIAM J. Appl. Math. 15 (6), pp. 1400–1412.

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External Links:

Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

(9.13.1), (9.13.10), (9.13.11), (9.13.12), (9.13.2), (9.13.3), (9.13.4), (9.13.5), (9.13.6), (9.13.7), (9.13.8), (9.13.9), §9.13(i), §9.13(i).

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18: 35.9 Applications

… ►See James (1964), Muirhead (1982), Takemura (1984), Farrell (1985), and Chikuse (2003) for extensive treatments. … ►In the nascent area of applications of zonal polynomials to the limiting probability distributions of symmetric random matrices, one of the most comprehensive accounts is Rains (1998).

19: Bibliography M

… ►

Symmetric Functions and Hall Polynomials. 2nd edition, The Clarendon Press, Oxford University Press, New York-Oxford.

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Notes:

With contributions by A. Zelevinsky. Paperback version published by Oxford University Press, July 1999.

External Links:

ISBN 0-19-853489-2; 0-19-850450-0, MathReview (John R. Stembridge), ZentralBlatt

Referenced by:

§18.37(iii), §26.19, §35.4(i), §35.4(ii).

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New ladder operators for a rational extension of the harmonic oscillator and superintegrability of some two-dimensional systems. J. Math. Phys. 54 (10), pp. Paper 102102, 12 pp..

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External Links:

MathReview Entry

Referenced by:

§18.36(vi).

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Connection between quantum systems involving the fourth Painlevé transcendent and $k$-step rational extensions of the harmonic oscillator related to Hermite exceptional orthogonal polynomial. J. Math. Phys. 57 (5), pp. Paper 052101, 15 pp..

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External Links:

ISSN 0022-2488, Document, Link, MathReview Entry

Referenced by:

§18.38(iii).

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Lattice Statistics. In Applied Combinatorial Mathematics, E. F. Beckenbach (Ed.), University of California Engineering and Physical Sciences Extension Series, pp. 96–143.

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Notes:

Reprinted by Robert E. Krieger Publishing Co. in 1981.

External Links:

ZentralBlatt

Referenced by:

§26.20.

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… ►

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Notes:

Mpmath is a pure-Python library for multiprecision floating-point arithmetic. It provides an extensive set of transcendental functions, unlimited exponent sizes, complex numbers, interval arithmetic, numerical integration and differentiation, root-finding, linear algebra, and much more.

External Links:

mpmath

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.

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20: Foreword

… ►The provision of standard reference data of this type is a core function of NIST. … ►The new printed volume, the NIST Handbook of Mathematical Functions, serves a similar function as the original A&S, though it is heavily updated and extended. The online version, the NIST Digital Library of Mathematical Functions (DLMF), presents the same technical information along with extensions and innovative interactive features consistent with the new medium. The DLMF may well serve as a model for the effective presentation of highly mathematical reference material on the Web. ►The production of these new resources has been a very complex undertaking some 10 years in the making. …