continuous%20q-Hermite%20polynomials
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21: 28 Mathieu Functions and Hill’s Equation
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22: 18.22 Hahn Class: Recurrence Relations and Differences
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§18.22(i) Recurrence Relations in
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… ►§18.22(iii) -Differences
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…23: 18.20 Hahn Class: Explicit Representations
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§18.20(i) Rodrigues Formulas
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… ►§18.20(ii) Hypergeometric Function and Generalized Hypergeometric Functions
… ►(For symmetry properties of with respect to , , , see Andrews et al. (1999, Corollary 3.3.4).) …24: William P. Reinhardt
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►Reinhardt is a theoretical chemist and atomic physicist, who has always been interested in orthogonal polynomials and in the analyticity properties of the functions of mathematical physics.
…Older work on the scattering theory of the atomic Coulomb problem led to the discovery of new classes of orthogonal polynomials relating to the spectral theory of Schrödinger operators, and new uses of old ones: this work was strongly motivated by his original ownership of a 1964 hard copy printing of the original AMS 55 NBS Handbook of Mathematical Functions.
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►In November 2015, Reinhardt was named Senior Associate Editor of the DLMF and Associate Editor for Chapters 20, 22, and 23.
25: 8.26 Tables
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Khamis (1965) tabulates for , to 10D.
Abramowitz and Stegun (1964, pp. 245–248) tabulates for , to 7D; also for , to 6S.
Pagurova (1961) tabulates for , to 4-9S; for , to 7D; for , to 7S or 7D.
Zhang and Jin (1996, Table 19.1) tabulates for , to 7D or 8S.
26: 23 Weierstrass Elliptic and Modular
Functions
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27: 3.4 Differentiation
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►If is continuous on the interval defined in §3.3(i), then the remainder in (3.4.1) is given by
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28: 1.17 Integral and Series Representations of the Dirac Delta
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►From the mathematical standpoint the left-hand side of (1.17.2) can be interpreted as a generalized integral in the sense that
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►for all functions that are continuous when , and for each , converges absolutely for all sufficiently large values of .
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►More generally, assume is piecewise continuous (§1.4(ii)) when for any finite positive real value of , and for each , converges absolutely for all sufficiently large values of .
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►provided that is continuous when , and for each , converges absolutely for all sufficiently large values of (as in the case of (1.17.6)).
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►provided that is continuous and of period ; see §1.8(ii).
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29: 18.39 Applications in the Physical Sciences
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►The properties of determine whether the spectrum, this being the set of eigenvalues of , is discrete, continuous, or mixed, see §1.18.
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►Such a superposition yields continuous time evolution of the probability density .
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►Derivations of (18.39.42) appear in Bethe and Salpeter (1957, pp. 12–20), and Pauling and Wilson (1985, Chapter V and Appendix VII), where the derivations are based on (18.39.36), and is also the notation of Piela (2014, §4.7), typifying the common use of the associated Coulomb–Laguerre polynomials in theoretical quantum chemistry.
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►For these are the repulsive CP OP’s with corresponding to the continuous spectrum of , , and for we have the attractive CP OP’s, where the spectrum is complemented by the infinite set of bound state eigenvalues for fixed .
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►Given that in both the attractive and repulsive cases, the expression for the absolutely continuous, , part of the function of (18.35.6) may be simplified:
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30: 18.40 Methods of Computation
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