symbolic operations
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11: Bibliography R
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Methods of Modern Mathematical Physics, Vol. 4, Analysis of Operators.
Academic Press, New York.
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On the computation of Lamé functions, of eigenvalues and eigenfunctions of some potential operators.
Z. Angew. Math. Mech. 78 (1), pp. 66–72.
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On the foundations of combinatorial theory. VIII. Finite operator calculus.
J. Math. Anal. Appl. 42, pp. 684–760.
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On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media.
In Differential Operators and Related Topics, Vol. I (Odessa,
1997),
Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.
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12: 2.1 Definitions and Elementary Properties
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§2.1(i) Asymptotic and Order Symbols
… ►§2.1(ii) Integration and Differentiation
… ►Symbolically, … ►Most operations on asymptotic expansions can be carried out in exactly the same manner as for convergent power series. … ►Symbolically, …13: Errata
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►This especially included updated information on matrix analysis, measure theory, spectral analysis, and a new section on linear second order differential operators and eigenfunction expansions.
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►The specific updates to Chapter 1 include the addition of an entirely new subsection §1.18 entitled “Linear Second Order Differential Operators and Eigenfunction Expansions” which is a survey of the formal spectral analysis of second order differential operators.
The spectral theory of these operators, based on Sturm-Liouville and Liouville normal forms, distribution theory, is now discussed more completely, including linear algebra, matrices, matrices as linear operators, orthonormal expansions, Stieltjes integrals/measures, generating functions.
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Usability
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Subsections 1.15(vi), 1.15(vii), 2.6(iii)
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Linkage of mathematical symbols to their definitions were corrected or improved.
A number of changes were made with regard to fractional integrals and derivatives. In §1.15(vi) a reference to Miller and Ross (1993) was added, the fractional integral operator of order was more precisely identified as the Riemann-Liouville fractional integral operator of order , and a paragraph was added below (1.15.50) to generalize (1.15.47). In §1.15(vii) the sentence defining the fractional derivative was clarified. In §2.6(iii) the identification of the Riemann-Liouville fractional integral operator was made consistent with §1.15(vi).
14: 18.1 Notation
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-Differences
►Forward differences: … ►Backward differences: … ►Central differences in imaginary direction: … ►-Pochhammer Symbol
…15: DLMF Project News
error generating summary16: 18.22 Hahn Class: Recurrence Relations and Differences
17: Bibliography M
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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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The supports of measures associated with orthogonal polynomials and the spectra of the related selfadjoint operators.
Rocky Mountain J. Math. 21 (1), pp. 501–527.
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Recursion relations for the -
symbols.
Nuclear Physics A 113 (1), pp. 215–220.
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SciFace Software, Paderborn, Germany.
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18: Bibliography D
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Computing spectra of linear operators using the Floquet-Fourier-Hill method.
J. Comput. Phys. 219 (1), pp. 296–321.
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Texas Instruments, Inc..
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Linear operators. Part II.
Wiley Classics Library, John Wiley & Sons, Inc., New York.
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Differential-difference operators associated to reflection groups.
Trans. Amer. Math. Soc. 311 (1), pp. 167–183.
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Novel identities for simple -
symbols.
J. Mathematical Phys. 16, pp. 318–319.
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19: 18.20 Hahn Class: Explicit Representations
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►For comments on the use of the forward-difference operator
, the backward-difference operator
, and the central-difference operator
, see §18.2(ii).
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18.20.9
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18.20.10