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31: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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Bounded and Unbounded Linear Operators

… ► … ► … ►Now formulas (1.18.13)–(1.18.20) apply. … ►Now formulas (1.18.13)–(1.18.20) apply. …

32: Bibliography N

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On simplified asymptotic formulas for a class of Mathieu functions. SIAM J. Math. Anal. 15 (6), pp. 1205–1213.

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External Links:

ISSN 0036-1410, Document, MathReview (Harold Exton), ZentralBlatt

Referenced by:

§28.26(ii).

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On a simplified asymptotic formula for the Mathieu function of the third kind. SIAM J. Math. Anal. 18 (6), pp. 1616–1629.

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ISSN 0036-1410, Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§28.26(ii).

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An explicit formula for the coefficients in Laplace’s method. Constr. Approx. 38 (3), pp. 471–487.

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ISSN 0176-4276, Document, Link, MathReview (Pedro J. Pagola), ZentralBlatt

Referenced by:

§5.11(i), §5.11(i).

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Generalization of Binet’s Gamma function formulas. Integral Transforms Spec. Funct. 24 (8), pp. 597–606.

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ISSN 1065-2469, Document, FullText, MathReview (Daniele Ritelli), ZentralBlatt

Referenced by:

§5.11(i), §5.11(i).

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Uniform Asymptotic Approximations of Solutions of Second-order Linear Differential Equations, with a Coalescing Simple Turning Point and Simple Pole. Ph.D. Thesis, University of Maryland, College Park, MD.

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Referenced by:

§2.8(vi).

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33: Bibliography B

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Transcendental Functions Satisfying Nonhomogeneous Linear Differential Equations. The Macmillan Co., New York.

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External Links:

MathReview (H. Hochstadt), ZentralBlatt

Referenced by:

§11.4(i), §11.4(v), §11.5(ii), §11.5(i), §11.5(ii), §11.5(iii), §11.7(ii), §11.7(iii), §11.7(v), §11.9(i), §11.9(iv), §11.9(iv), Ch.11, §14.29.

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Mathieu functions of general order: Connection formulae, base functions and asymptotic formulae. I–V. Philos. Trans. Roy. Soc. London Ser. A 301, pp. 75–162.

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ISSN 0080-4614, FullText, MathReview (J. Meixner), ZentralBlatt

Referenced by:

§28.8(iv).

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An Introduction to Linear Difference Equations. Dover Publications Inc., New York.

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Notes:

Unaltered republication of the original 1927 Harvard University edition.

External Links:

MathReview, ZentralBlatt

Referenced by:

§8.1, Notations P, Notations Q.

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Discrete ordinate solution of Fokker-Planck equations with non-linear coefficients. Phys. Rev. A 31 (3), pp. 1855–1868.

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Referenced by:

§18.39(iii).

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Error estimates for the solution of linear systems. SIAM J. Sci. Comput. 21 (2), pp. 764–781.

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ISSN 1095-7197, Document, MathReview (Dimitrios Noutsos), ZentralBlatt

Referenced by:

§3.2(iii).

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34: Bibliography R

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A non-negative representation of the linearization coefficients of the product of Jacobi polynomials. Canad. J. Math. 33 (4), pp. 915–928.

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ISSN 0008-414X, Document, Link, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.18(v).

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Fourier analysis and signal processing by use of the Möbius inversion formula. IEEE Trans. Acoustics, Speech, Signal Processing 38, pp. 458–470.

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External Links:

ZentralBlatt

Referenced by:

§27.17.

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General Computation Methods of Chebyshev Approximation. The Problems with Linear Real Parameters. Publishing House of the Academy of Science of the Ukrainian SSR, Kiev.

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Notes:

English translation, AEC-TR-4491, United States Atomic Energy Commission

External Links:

MathReview (A. S. Householder)

Referenced by:

§3.11(iii).

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Another proof of the triple sum formula for Wigner $9j$-symbols. J. Math. Phys. 40 (12), pp. 6689–6691.

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ISSN 0022-2488, Document, MathReview (Michael Wüstner), ZentralBlatt

Referenced by:

§34.6.

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35: Bibliography M

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A determinant formula for a class of rational solutions of Painlevé V equation. Nagoya Math. J. 168, pp. 1–25.

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ISSN 0027-7630, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.8(v).

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On a class of algebraic solutions to the Painlevé VI equation, its determinant formula and coalescence cascade. Funkcial. Ekvac. 46 (1), pp. 121–171.

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ISSN 0532-8721, FullText, MathReview (Andrew Pickering), ZentralBlatt

Referenced by:

§32.9(iii).

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On the choice of standard solutions for a homogeneous linear differential equation of the second order. Quart. J. Mech. Appl. Math. 3 (2), pp. 225–235.

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External Links:

Document, MathReview (W. R. Wasow), ZentralBlatt

Referenced by:

§12.2(vi), §2.7(iv), §2.7(iv).

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Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions. Ramanujan J. 6 (1), pp. 7–149.

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External Links:

Document, ZentralBlatt

Referenced by:

§27.13(iv).

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The $q$-analogue of Stirling’s formula. Rocky Mountain J. Math. 14 (2), pp. 403–413.

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External Links:

ISSN 0035-7596, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§5.18(ii).

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36: 1.16 Distributions

… ►The linear space of all test functions with the above definition of convergence is called a test function space. … ►A mapping $\mathrm{\Lambda }:𝒟(I)\to ℂ$ is a linear functional if … ►A tempered distribution is a continuous linear functional $\mathrm{\Lambda }$ on $𝒯$. … ►A distribution in ${ℝ}^n$ is a continuous linear functional on ${𝒟}_n$. … ►Tempered distributions are continuous linear functionals on this space of test functions. …

37: 18.27 $q$-Hahn Class

… ►The $q$-hypergeometric OP’s comprise the $q$-Hahn class (or $q$-linear lattice class) OP’s and the Askey–Wilson class (or $q$-quadratic lattice class) OP’s (§18.28). … ►For other formulas, including $q$-difference equations, recurrence relations, duality formulas, special cases, and limit relations, see Koekoek et al. (2010, Chapter 14). …