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1: 28.2 Definitions and Basic Properties
§28.2(vi) Eigenfunctions
2: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
The analogous orthonormality is … This may be compared to (1.17.21), the resulting Fourier, or eigenfunction, expansion … and completeness relation It is to be noted that if any of the λ 𝝈 have degenerate sub-spaces, that is subspaces of orthogonal eigenfunctions with identical eigenvalues, that in the expansions below all such distinct eigenfunctions are to be included. …
3: 18.28 Askey–Wilson Class
y ) such that P n ( z ) = p n ( 1 2 ( z + z 1 ) ) in the Askey–Wilson case, and P n ( y ) = p n ( q y + c q y + 1 ) in the q -Racah case, and both are eigenfunctions of a second order q -difference operator similar to (18.27.1). …
§18.28(ii) Askey–Wilson Polynomials
Recurrence Relation
§18.28(viii) q -Racah Polynomials
Genest et al. (2016) showed that these polynomials coincide with the nonsymmetric Wilson polynomials in Groenevelt (2007).
4: 1.17 Integral and Series Representations of the Dirac Delta
subject to certain conditions on the function ϕ ( x ) . …
1.17.6 lim n n π e n ( x a ) 2 ϕ ( x ) d x = ϕ ( a ) ,
The last condition is satisfied, for example, when ϕ ( x ) = O ( e α x 2 ) as x ± , where α is a real constant. … In the language of physics and applied mathematics, these equations indicate the normalizations chosen for these non- L 2 improper eigenfunctions of the differential operators (with derivatives respect to spatial co-ordinates) which generate them; the normalizations (1.17.12_1) and (1.17.12_2) are explicitly derived in Friedman (1990, Ch. 4), the others follow similarly. Equations (1.17.12_1) through (1.17.16) may re-interpreted as spectral representations of completeness relations, expressed in terms of Dirac delta distributions, as discussed in §1.18(v), and §1.18(vi) Further mathematical underpinnings are referenced in §1.17(iv). …
5: Errata
  • Chapter 1 Additions

    The following additions were made in Chapter 1:

  • Chapter 18 Additions

    The following additions were made in Chapter 18:

    • §18.2

      In Subsection 18.2(i), Equation (18.2.1_5); the paragraph title “Orthogonality on Finite Point Sets” has been changed to “Orthogonality on Countable Sets”, and there are minor changes in the presentation of the final paragraph, including a new equation (18.2.4_5). The presentation of Subsection 18.2(iii) has changed, Equation (18.2.5_5) was added and an extra paragraph on standardizations has been included. The presentation of Subsection 18.2(iv) has changed and it has been expanded with two extra paragraphs and several new equations, (18.2.9_5), (18.2.11_1)–(18.2.11_9). Subsections 18.2(v) (with (18.2.12_5), (18.2.14)–(18.2.17)) and 18.2(vi) (with (18.2.17)–(18.2.20)) have been expanded. New subsections, 18.2(vii)18.2(xii), with Equations (18.2.21)–(18.2.46),

    • §18.3

      A new introduction, minor changes in the presentation, and three new paragraphs.

    • §18.5

      Extra details for Chebyshev polynomials, and Equations (18.5.4_5), (18.5.11_1)–(18.5.11_4), (18.5.17_5).

    • §18.8

      Line numbers and two extra rows were added to Table 18.8.1.

    • §18.9

      Subsection 18.9(i) has been expanded, and 18.9(iii) has some additional explanation. Equations (18.9.2_1), (18.9.2_2), (18.9.18_5) and Table 18.9.2 were added.

    • §18.12

      Three extra generating functions, (18.12.2_5), (18.12.3_5), (18.12.17).

    • §18.14

      Equation (18.14.3_5). New subsection, 18.14(iv), with Equations (18.14.25)–(18.14.27).

    • §18.15

      Equation (18.15.4_5).

    • §18.16

      The title of Subsection 18.16(iii) was changed from “Ultraspherical and Legendre” to “Ultraspherical, Legendre and Chebyshev”. New subsection, 18.16(vii) Discriminants, with Equations (18.16.19)–(18.16.21).

    • §18.17

      Extra explanatory text at many places and seven extra integrals (18.17.16_5), (18.17.21_1)–(18.17.21_3), (18.17.28_5), (18.17.34_5), (18.17.41_5).

    • §18.18

      Extra explanatory text at several places and the title of Subsection 18.18(iv) was changed from “Connection Formulas” to “Connection and Inversion Formulas”.

    • §18.19

      A new introduction.

    • §18.21

      Equation (18.21.13).

    • §18.25

      Extra explanatory text in Subsection 18.25(i) and the title of Subsection 18.25(ii) was changed from “Weights and Normalizations: Continuous Cases” to “Weights and Standardizations: Continuous Cases”.

    • §18.26

      In Subsection 18.26(i) an extra paragraph on dualities has been included, with Equations (18.26.4_1), (18.26.4_2).

    • §18.27

      Extra text at the start of this section and twenty seven extra formulas, (18.27.4_1), (18.27.4_2), (18.27.6_5), (18.27.9_5), (18.27.12_5), (18.27.14_1)–(18.27.14_6), (18.27.17_1)–(18.27.17_3), (18.27.20_5), (18.27.25), (18.27.26), (18.28.1_5).

    • §18.28

      A big expansion. Six extra formulas in Subsection 18.28(ii) ((18.28.6_1)–(18.28.6_5)) and three extra formulas in Subsection 18.28(viii) ((18.28.21)–(18.28.23)). New subsections, 18.28(ix)18.28(xi), with Equations (18.28.23)–(18.28.34).

    • §18.30

      Originally this section did not have subsections. The original seven formulas have now more explanatory text and are split over two subsections. New subsections 18.30(iii)18.30(viii), with Equations (18.30.8)–(18.30.31).

    • §18.32

      This short section has been expanded, with Equation (18.32.2).

    • §18.33

      Additional references and a new large subsection, 18.33(vi), including Equations (18.33.17)–(18.33.32).

    • §18.34

      This section has been expanded, including an extra orthogonality relations (18.34.5_5), (18.34.7_1)–(18.34.7_3).

    • §18.35

      This section on Pollaczek polynomials has been significantly updated with much more explanations and as well to include the Pollaczek polynomials of type 3 which are the most general with three free parameters. The Pollaczek polynomials which were previously treated, namely those of type 1 and type 2 are special cases of the type 3 Pollaczek polynomials. In the first paragraph of this section an extensive description of the relations between the three types of Pollaczek polynomials is given which was lacking previously. Equations (18.35.0_5), (18.35.2_1)–(18.35.2_5), (18.35.4_5), (18.35.6_1)–(18.35.6_6), (18.35.10).

    • §18.36

      This section on miscellaneous polynomials has been expanded with new subsections, 18.36(v) on non-classical Laguerre polynomials and 18.36(vi) with examples of exceptional orthogonal polynomials, with Equations (18.36.1)–(18.36.10). In the titles of Subsections 18.36(ii) and 18.36(iii) we replaced “OP’s” by “Orthogonal Polynomials”.

    • §18.38

      The paragraphs of Subsection 18.38(i) have been re-ordered and one paragraph was added. The title of Subsection 18.38(ii) was changed from “Classical OP’s: Other Applications” to “Classical OP’s: Mathematical Developments and Applications”. Subsection 18.38(iii) has been expanded with seven new paragraphs, and Equations (18.38.4)–(18.38.11).

    • §18.39

      This section was completely rewritten. The previous 18.39(i) Quantum Mechanics has been replaced by Subsections 18.39(i) Quantum Mechanics and 18.39(ii) A 3D Separable Quantum System, the Hydrogen Atom, containing the same essential information; the original content of the subsection is reproduced below for reference. Subsection 18.39(ii) was moved to 18.39(v) Other Applications. New subsections, 18.39(iii) Non Classical Weight Functions of Utility in DVR Method in the Physical Sciences, 18.39(iv) Coulomb–Pollaczek Polynomials and J-Matrix Methods; Equations (18.39.7)–(18.39.48); and Figures 18.39.1, 18.39.2.

      The original text of 18.39(i) Quantum Mechanics was:

      “Classical OP’s appear when the time-dependent Schrödinger equation is solved by separation of variables. Consider, for example, the one-dimensional form of this equation for a particle of mass m with potential energy V ( x ) :

      errata.1 ( 2 2 m 2 x 2 + V ( x ) ) ψ ( x , t ) = i t ψ ( x , t ) ,

      where is the reduced Planck’s constant. On substituting ψ ( x , t ) = η ( x ) ζ ( t ) , we obtain two ordinary differential equations, each of which involve the same constant E . The equation for η ( x ) is

      errata.2 d 2 η d x 2 + 2 m 2 ( E V ( x ) ) η = 0 .

      For a harmonic oscillator, the potential energy is given by

      errata.3 V ( x ) = 1 2 m ω 2 x 2 ,

      where ω is the angular frequency. For (18.39.2) to have a nontrivial bounded solution in the interval < x < , the constant E (the total energy of the particle) must satisfy

      errata.4 E = E n = ( n + 1 2 ) ω , n = 0 , 1 , 2 , .

      The corresponding eigenfunctions are

      errata.5 η n ( x ) = π 1 4 2 1 2 n ( n ! b ) 1 2 H n ( x / b ) e x 2 / 2 b 2 ,

      where b = ( / m ω ) 1 / 2 , and H n is the Hermite polynomial. For further details, see Seaborn (1991, p. 224) or Nikiforov and Uvarov (1988, pp. 71-72).

      A second example is provided by the three-dimensional time-independent Schrödinger equation

      errata.6 2 ψ + 2 m 2 ( E V ( 𝐱 ) ) ψ = 0 ,

      when this is solved by separation of variables in spherical coordinates (§1.5(ii)). The eigenfunctions of one of the separated ordinary differential equations are Legendre polynomials. See Seaborn (1991, pp. 69-75).

      For a third example, one in which the eigenfunctions are Laguerre polynomials, see Seaborn (1991, pp. 87-93) and Nikiforov and Uvarov (1988, pp. 76-80 and 320-323).”

    • Section 18.40

      The old section is now Subsection 18.40(i) and a large new subsection, 18.40(ii), on the classical moment problem has been added, with formulae (18.40.1)–(18.40.10) and Figures 18.40.1, 18.40.2.

  • 6: 18.38 Mathematical Applications
    The Askey–Gasper inequalitySUSY leads to algebraic simplifications in generating excited states, and partner potentials with closely related energy spectra, from knowledge of a single ground state wave function. …
    7: 31.17 Physical Applications
    for the common eigenfunction Ψ ( 𝐱 ) = Ψ ( x s , x t , x u ) , where a is the coupling parameter of interacting spins. …The operators 𝐉 2 and 𝐻 s admit separation of variables in z 1 , z 2 , leading to the following factorization of the eigenfunction Ψ ( 𝐱 ) :
    31.17.4 Ψ ( 𝐱 ) = ( z 1 z 2 ) s 1 4 ( ( z 1 1 ) ( z 2 1 ) ) t 1 4 ( ( z 1 a ) ( z 2 a ) ) u 1 4 w ( z 1 ) w ( z 2 ) ,
    For more details about the method of separation of variables and relation to special functions see Olevskiĭ (1950), Kalnins et al. (1976), Miller (1977), and Kalnins (1986). …
    8: 18.27 q -Hahn Class
    Thus in addition to a relation of the form (18.27.2), such systems may also satisfy orthogonality relations with respect to a continuous weight function on some interval. …
    From Big q -Jacobi to Jacobi
    From Big q -Jacobi to Little q -Jacobi
    From Little q -Jacobi to Jacobi
    From Little q -Laguerre to Laguerre
    9: 18.39 Applications in the Physical Sciences
    While non-normalizable continuum, or scattering, states are mentioned, with appropriate references in what follows, focus is on the L 2 eigenfunctions corresponding to the point, or discrete, spectrum, and representing bound rather than scattering states, these former being expressed in terms of OP’s or EOP’s. … which in one dimensional systems are typically non-degenerate, namely there is only a single eigenfunction corresponding to each ϵ n , n 0 . … Namely the k th eigenfunction, listed in order of increasing eigenvalues, starting at k = 0 , has precisely k nodes, as real zeros of wave-functions away from boundaries are often referred to. … The fact that both the eigenvalues of (18.39.31) and the scaling of the r co-ordinate in the eigenfunctions, (18.39.30), depend on the sum p + l + 1 leads to the substitution … As the scattering eigenfunctions of Chapter 33, are not OP’s, their further discussion is deferred to §18.39(iv), where discretized representations of these scattering states are introduced, Laguerre and Pollaczek OP’s then playing a key role. …
    10: Bibliography V
  • G. Valent (1986) An integral transform involving Heun functions and a related eigenvalue problem. SIAM J. Math. Anal. 17 (3), pp. 688–703.
  • O. Vallée and M. Soares (2010) Airy Functions and Applications to Physics. Second edition, Imperial College Press, London.
  • A. J. van der Poorten (1980) Some Wonderful Formulas an Introduction to Polylogarithms. In Proceedings of the Queen’s Number Theory Conference, 1979 (Kingston, Ont., 1979), R. Ribenboim (Ed.), Queen’s Papers in Pure and Appl. Math., Vol. 54, Kingston, Ont., pp. 269–286.
  • H. Volkmer (1982) Integral relations for Lamé functions. SIAM J. Math. Anal. 13 (6), pp. 978–987.
  • H. Volkmer (2004a) Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.