with other orthogonal polynomials
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21: 18.29 Asymptotic Approximations for -Hahn and Askey–Wilson Classes
§18.29 Asymptotic Approximations for -Hahn and Askey–Wilson Classes
►Ismail (1986) gives asymptotic expansions as , with and other parameters fixed, for continuous -ultraspherical, big and little -Jacobi, and Askey–Wilson polynomials. …For Askey–Wilson the leading term is given by … ►For a uniform asymptotic expansion of the Stieltjes–Wigert polynomials, see Wang and Wong (2006). ►For asymptotic approximations to the largest zeros of the -Laguerre and continuous -Hermite polynomials see Chen and Ismail (1998).22: 18.1 Notation
23: 18.39 Applications in the Physical Sciences
Other Analytically Solved Schrödinger Equations
… ►The Coulomb–Pollaczek Polynomials
… ►These cases correspond to the two distinct orthogonality conditions of (18.35.6) and (18.35.6_3). … ►§18.39(v) Other Applications
… ►For interpretations of zeros of classical OP’s as equilibrium positions of charges in electrostatic problems (assuming logarithmic interaction), see Ismail (2000a, b).24: 18.2 General Orthogonal Polynomials
25: 18.36 Miscellaneous Polynomials
§18.36(ii) Sobolev Orthogonal Polynomials
… ►§18.36(iii) Multiple Orthogonal Polynomials
►These are polynomials in one variable that are orthogonal with respect to a number of different measures. … ►§18.36(iv) Orthogonal Matrix Polynomials
… ►§18.36(vi) Exceptional Orthogonal Polynomials
…26: 18.9 Recurrence Relations and Derivatives
§18.9(i) Recurrence Relations
… ►For the other classical OP’s see Table 18.9.2. … ►§18.9(ii) Contiguous Relations in the Parameters and the Degree
… ►§18.9(iii) Derivatives
►Jacobi
…27: 31.11 Expansions in Series of Hypergeometric Functions
28: 18.15 Asymptotic Approximations
§18.15 Asymptotic Approximations
►§18.15(i) Jacobi
… ►§18.15(ii) Ultraspherical
… ►§18.15(vi) Other Approximations
… ►See also Dunster (1999), Atia et al. (2014) and Temme (2015, Chapter 32).29: 18.34 Bessel Polynomials
§18.34(i) Definitions and Recurrence Relation
… ►§18.34(ii) Orthogonality
… ►Hence the full system of polynomials cannot be orthogonal on the line with respect to a positive weight function, but this is possible for a finite system of such polynomials, the Romanovski–Bessel polynomials, if : … ►§18.34(iii) Other Properties
… ►In this limit the finite system of Jacobi polynomials which is orthogonal on (see §18.3) tends to the finite system of Romanovski–Bessel polynomials which is orthogonal on (see (18.34.5_5)). …30: Errata
The following additions were made in Chapter 18:
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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),
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Section 18.3
A new introduction, minor changes in the presentation, and three new paragraphs.
- Section 18.5
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Section 18.8
Line numbers and two extra rows were added to Table 18.8.1.
- Section 18.9
- Section 18.12
- Section 18.14
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Section 18.15
Equation (18.15.4_5).
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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).
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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).
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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”.
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Section 18.19
A new introduction.
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Section 18.21
Equation (18.21.13).
- Section 18.25
- Section 18.26
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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
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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).
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Section 18.32
This short section has been expanded, with Equation (18.32.2).
- Section 18.33
- Section 18.34
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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).
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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”.
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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).
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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 with potential energy :
errata.1where is the reduced Planck’s constant. On substituting , we obtain two ordinary differential equations, each of which involve the same constant . The equation for is
errata.2For a harmonic oscillator, the potential energy is given by
errata.3where is the angular frequency. For (18.39.2) to have a nontrivial bounded solution in the interval , the constant (the total energy of the particle) must satisfy
errata.4 .The corresponding eigenfunctions are
errata.5where , and 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.6when 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