continuous q-Hermite polynomials
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21: 24.17 Mathematical Applications
§24.17 Mathematical Applications
… ►Let denote the class of functions that have continuous derivatives on and are polynomials of degree at most in each interval , . … ►§24.17(iii) Number Theory
►Bernoulli and Euler numbers and polynomials occur in: number theory via (24.4.7), (24.4.8), and other identities involving sums of powers; the Riemann zeta function and -series (§25.15, Apostol (1976), and Ireland and Rosen (1990)); arithmetic of cyclotomic fields and the classical theory of Fermat’s last theorem (Ribenboim (1979) and Washington (1997)); -adic analysis (Koblitz (1984, Chapter 2)).22: 1.4 Calculus of One Variable
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§1.4(ii) Continuity
… ► … ► … ►For an example, see Figure 1.4.1 … ►Absolutely Continuous Stieltjes Measure
…23: 18.38 Mathematical Applications
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Approximation Theory
… ►The terminology DVR arises as an otherwise continuous variable, such as the co-ordinate , is replaced by its values at a finite set of zeros of appropriate OP’s resulting in expansions using functions localized at these points. … ►Integrable Systems
… ►Ultraspherical polynomials are zonal spherical harmonics. … ►Group Representations
…24: Bibliography J
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Fonctions de Mathieu et polynômes de Klein-Gordon.
C. R. Acad. Sci. Paris Sér. I Math. 325 (7), pp. 713–716 (French).
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Uniform asymptotic expansions for Meixner polynomials.
Constr. Approx. 14 (1), pp. 113–150.
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Asymptotic formulas for the zeros of the Meixner polynomials.
J. Approx. Theory 96 (2), pp. 281–300.
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Continuous Univariate Distributions.
2nd edition, Vol. I, John Wiley & Sons Inc., New York.
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Continuous Univariate Distributions.
2nd edition, Vol. II, John Wiley & Sons Inc., New York.
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25: Bibliography F
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Multivariate Calculation. Use of the Continuous Groups.
Springer Series in Statistics, Springer-Verlag, New York.
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Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. II.
J. Math. Anal. Appl. 7 (3), pp. 440–451.
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Asymptotic expansions of a class of hypergeometric polynomials with respect to the order.
J. Math. Anal. Appl. 6 (3), pp. 394–403.
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From continuous to discrete Painlevé equations.
J. Math. Anal. Appl. 180 (2), pp. 342–360.
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Continuous and Discrete Painlevé Equations.
In Painlevé Transcendents: Their Asymptotics and Physical Applications, D. Levi and P. Winternitz (Eds.),
NATO Adv. Sci. Inst. Ser. B Phys., Vol. 278, pp. 33–47.
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26: 1.10 Functions of a Complex Variable
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►Also, let be analytic within , continuous within and on , and real on .
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►If is analytic within a simple closed contour , and continuous within and on —except in both instances for a finite number of singularities within —then
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►If is continuous on and analytic in , then attains its maximum on .
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►Moreover, if is bounded and is continuous on and harmonic in , then is maximum at some point on .
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►(b) By specifying the value of at a point (not a branch point), and requiring to be continuous on any path that begins at and does not pass through any branch points or other singularities of .
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27: 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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The Coulomb–Pollaczek Polynomials
… ►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 . … ►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: …28: 3.11 Approximation Techniques
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§3.11(i) Minimax Polynomial Approximations
►Let be continuous on a closed interval . … ►Assume that is continuous on and let , , and be the zeros of in arranged so that … ►Furthermore, if , then the convergence of (3.11.11) is usually very rapid; compare (1.8.7) with arbitrary. … ►Let be continuous on a closed interval and be a continuous nonvanishing function on : is called a weight function. …29: 18.2 General Orthogonal Polynomials
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►Here is continuous or piecewise continuous or integrable such that
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►This happens, for example, with the continuous Hahn polynomials and Meixner–Pollaczek polynomials (§18.20(i)).
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Kernel Polynomials
… ►The measure is not necessarily absolutely continuous (i. … ►Sheffer Polynomials
…30: Bibliography
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Some orthogonal -polynomials.
Math. Nachr. 30, pp. 47–61.
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Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials.
Mem. Amer. Math. Soc. 54 (319), pp. iv+55.
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Continuous Hahn polynomials.
J. Phys. A 18 (16), pp. L1017–L1019.
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Continuous
-Hermite Polynomials when
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In
-series and Partitions (Minneapolis, MN, 1988),
IMA Vol. Math. Appl., Vol. 18, pp. 151–158.
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Singular Continuous Spectrum for a Class of Almost Periodic Jacobi Matrices.
Bulletin of the American Mathematical Society 6 (1), pp. 81–85.
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