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11: 9.17 Methods of Computation
However, in the case of Ai ( z ) and Bi ( z ) this accuracy can be increased considerably by use of the exponentially-improved forms of expansion supplied in §9.7(v). … For details, including the application of a generalized form of Gaussian quadrature, see Gordon (1969, Appendix A) and Schulten et al. (1979). … The second method is to apply generalized Gauss–Laguerre quadrature (§3.5(v)) to the integral (9.5.8). …
12: 18.14 Inequalities
Laguerre
Laguerre
Laguerre
Laguerre
13: 18.9 Recurrence Relations and Derivatives
First Form
Second Form
Laguerre
Laguerre
Further n -th derivative formulas relating two different Laguerre polynomials can be obtained from §13.3(ii) by substitution of (13.6.19). …
14: Errata
  • Chapter 18 Additions

    The following additions were made in Chapter 18:

    • Section 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),

    • Section 18.3

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

    • Section 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).

    • Section 18.8

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

    • Section 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.

    • Section 18.12

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

    • Section 18.14

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

    • Section 18.15

      Equation (18.15.4_5).

    • Section 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).

    • Section 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).

    • Section 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”.

    • Section 18.19

      A new introduction.

    • Section 18.21

      Equation (18.21.13).

    • Section 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”.

    • Section 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).

    • Section 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).

    • Section 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).

    • Section 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).

    • Section 18.32

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

    • Section 18.33

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

    • Section 18.34

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

    • Section 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).

    • Section 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”.

    • Section 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).

    • Section 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.

  • Subsection 33.14(iv)

    Just below (33.14.9), the constraint described in the text “ < ( ϵ ) 1 / 2 when ϵ < 0 ,” was removed. In Equation (33.14.13), the constraint ϵ 1 , ϵ 2 > 0 was added. In the line immediately below (33.14.13), it was clarified that s ( ϵ , ; r ) is exp ( r / n ) times a polynomial in r / n , instead of simply a polynomial in r . In Equation (33.14.14), a second equality was added which relates ϕ n , ( r ) to Laguerre polynomials. A sentence was added immediately below (33.14.15) indicating that the functions ϕ n , , n = , + 1 , , do not form a complete orthonormal system.

  • 15: 33.14 Definitions and Basic Properties
    33.14.14 ϕ n , ( r ) = ( 1 ) + 1 + n ( 2 / n 3 ) 1 / 2 s ( 1 / n 2 , ; r ) = ( 1 ) + 1 + n n + 2 ( ( n 1 ) ! ( n + ) ! ) 1 / 2 ( 2 r ) + 1 e r / n L n 1 ( 2 + 1 ) ( 2 r / n )
    Note that the functions ϕ n , , n = , + 1 , , do not form a complete orthonormal system. …
    16: 18.36 Miscellaneous Polynomials
    Similar OP’s can also be constructed for the Laguerre polynomials; see Koornwinder (1984b, (4.8)). …
    §18.36(v) Non-Classical Laguerre Polynomials L n ( k ) ( x ) , k = 1 , 2
    For the Laguerre polynomials L n ( α ) ( x ) this requires, omitting all strictly positive factors, …
    Type I X 1 -Laguerre EOP’s
    The resulting EOP’s, L ^ n ( k ) ( x ) , n = 1 , 2 , satisfy …
    17: 18.27 q -Hahn Class
    Together they form the q -Askey scheme. …
    Little q -Laguerre polynomials
    From Little q -Laguerre to Laguerre
    §18.27(v) q -Laguerre Polynomials
    From q -Laguerre to Laguerre
    18: 18.34 Bessel Polynomials
    For the confluent hypergeometric function F 1 1 and the generalized hypergeometric function F 0 2 , the Laguerre polynomial L n ( α ) and the Whittaker function W κ , μ see §16.2(ii), §16.2(iv), (18.5.12), and (13.14.3), respectively.
    18.34.1 y n ( x ; a ) = F 0 2 ( n , n + a 1 ; x 2 ) = ( n + a 1 ) n ( x 2 ) n F 1 1 ( n 2 n a + 2 ; 2 x ) = n ! ( 1 2 x ) n L n ( 1 a 2 n ) ( 2 x 1 ) = ( 1 2 x ) 1 1 2 a e 1 / x W 1 1 2 a , 1 2 ( a 1 ) + n ( 2 x 1 ) .
    18.34.7_1 ϕ n ( x ; λ ) = e λ e x ( 2 λ e x ) λ 1 2 y n ( λ 1 e x ; 2 2 λ ) / n ! = ( 1 ) n e λ e x ( 2 λ e x ) λ n 1 2 L n ( 2 λ 2 n 1 ) ( 2 λ e x ) = ( 2 λ ) 1 2 e x / 2 W λ , n + 1 2 λ ( 2 λ e x ) / n ! , n = 0 , 1 , , N = λ 3 2 , λ > 1 2 ,
    expressed in terms of Romanovski–Bessel polynomials, Laguerre polynomials or Whittaker functions, we have …
    18.34.8 lim α P n ( α , a α 2 ) ( 1 + α x ) P n ( α , a α 2 ) ( 1 ) = y n ( x ; a ) .
    19: 18.10 Integral Representations
    Laguerre
    Table 18.10.1 gives contour integral representations of the form …for the Jacobi, Laguerre, and Hermite polynomials. …
    Table 18.10.1: Classical OP’s: contour integral representations (18.10.8).
    p n ( x ) g 0 ( x ) g 1 ( z , x ) g 2 ( z , x ) c Conditions
    Laguerre
    20: Bibliography D
  • C. de la Vallée Poussin (1896b) Recherches analytiques sur la théorie des nombres premiers. Deuxième partie. Les fonctions de Dirichlet et les nombres premiers de la forme linéaire M x + N . Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).
  • A. Deaño, E. J. Huertas, and F. Marcellán (2013) Strong and ratio asymptotics for Laguerre polynomials revisited. J. Math. Anal. Appl. 403 (2), pp. 477–486.
  • A. Decarreau, M.-Cl. Dumont-Lepage, P. Maroni, A. Robert, and A. Ronveaux (1978a) Formes canoniques des équations confluentes de l’équation de Heun. Ann. Soc. Sci. Bruxelles Sér. I 92 (1-2), pp. 53–78.
  • C. F. Dunkl (2003) A Laguerre polynomial orthogonality and the hydrogen atom. Anal. Appl. (Singap.) 1 (2), pp. 177–188.