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21: 18.35 Pollaczek Polynomials

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§18.35(ii) Orthogonality

… ► … ►Then … ►For type 3 orthogonality (18.35.5) generalizes to … ►These polynomials also occur in connection with the Coulomb problem, see §18.39(iv).

22: 16.7 Relations to Other Functions

§16.7 Relations to Other Functions

►For orthogonal polynomials see Chapter 18. …

23: Roelof Koekoek

… ►Koekoek is mainly a teacher of mathematics and has published a few papers on orthogonal polynomials. He is also author of the book Hypergeometric Orthogonal Polynomials and Their $q$-Analogues (with P. … ►

18 Orthogonal Polynomials

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24: 18.24 Hahn Class: Asymptotic Approximations

§18.24 Hahn Class: Asymptotic Approximations

… ►This expansion is uniformly valid in any compact $x$-interval on the real line and is in terms of parabolic cylinder functions. Corresponding approximations are included for the zeros of $P_n^{(\lambda )}(nx;\varphi )$. ►

Approximations in Terms of Laguerre Polynomials

… ►Similar approximations are included for Jacobi, Krawtchouk, and Meixner polynomials.

25: 8.23 Statistical Applications

§8.23 Statistical Applications

… ►The function ${\mathrm{B}}_x(a,b)$ and its normalization $I_x(a,b)$ play a similar role in statistics in connection with the beta distribution; see Johnson et al. (1995, pp. 210–275). …

26: Bibliography W

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Asymptotics of orthogonal polynomials via recurrence relations. Anal. Appl. (Singap.) 10 (2), pp. 215–235.

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External Links:

ISSN 0219-5305, Document, Link, MathReview (Li Zhong Peng), ZentralBlatt

Referenced by:

§2.9(iii), §2.9(iii).

Encodings:

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Hypergeometric Series, Recurrence Relations and Some New Orthogonal Polynomials. Ph.D. Thesis, University of Wisconsin, Madison, WI.

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Referenced by:

§16.4(iii), §16.4(iii), §16.4(iii).

Encodings:

BibTeX

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On the central connection problem for the double confluent Heun equation. Math. Nachr. 195, pp. 267–276.

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External Links:

ISSN 0025-584X, MathReview, ZentralBlatt

Referenced by:

§31.12.

Encodings:

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On the connection formulas of the fourth Painlevé transcendent. Anal. Appl. (Singap.) 7 (4), pp. 419–448.

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External Links:

ISSN 0219-5305, Document, MathReview, ZentralBlatt

Referenced by:

§32.11(v), §32.11(v), §32.13(ii).

Encodings:

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On the connection formulas of the third Painlevé transcendent. Discrete Contin. Dyn. Syst. 23 (1-2), pp. 541–560.

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External Links:

ISSN 1078-0947, Document, MathReview (Galina V. Filipuk), ZentralBlatt

Referenced by:

§32.13(ii).

Encodings:

BibTeX

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27: 18.21 Hahn Class: Interrelations

§18.21 Hahn Class: Interrelations

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§18.21(i) Dualities

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§18.21(ii) Limit Relations and Special Cases

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Hahn $\to$ Jacobi

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Meixner $\to$ Laguerre

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28: 18.8 Differential Equations

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Table 18.8.1: Classical OP’s: differential equations $A(x)f^{\prime \prime }(x)+B(x)f^{\prime }(x)+C(x)f(x)+{\lambda }_{n}f(x)=0$.

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#	$f(x)$	$A(x)$	$B(x)$	$C(x)$	${\lambda }_n$
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4	$C_n^{(\lambda )}\left(x\right)$	$1-x^2$	$-(2\lambda +1)x$	$0$	$n(n+2\lambda )$
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8	$L_n^{(\alpha )}\left(x\right)$	$x$	$\alpha +1-x$	$0$	$n$
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12	$H_n\left(x\right)$	$1$	$-2x$	$0$	$2n$
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14	${\mathrm{𝐻𝑒}}_n\left(x\right)$	$1$	$-x$	$0$	$n$

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29: Bibliography L

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The two-point connection problem for differential equations of the Heun class. Teoret. Mat. Fiz. 101 (3), pp. 360–368 (Russian).

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Notes:

In Russian. English translation: Theoret. and Math. Phys. 101(1994), no. 3, pp. 1413–1418 (1995)

External Links:

ISSN 0564-6162, Document, MathReview, ZentralBlatt

Referenced by:

§31.18.

Encodings:

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The central two-point connection problem for the Heun class of ODEs. J. Phys. A 31 (18), pp. 4249–4261.

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External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§31.12.

Encodings:

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A partition function connected with the modulus five. Duke Math. J. 8 (4), pp. 631–655.

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External Links:

Document, MathReview (R. D. James), ZentralBlatt

Referenced by:

§17.18.

Encodings:

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Orthogonal polynomials, duality and association schemes. SIAM J. Math. Anal. 13 (4), pp. 656–663.

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External Links:

ISSN 0036-1410, Document, MathReview (C. L. Parihar), ZentralBlatt

Referenced by:

§18.28(viii), §18.28(xi), §18.38(iii).

Encodings:

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Approximation of orthogonal polynomials in terms of Hermite polynomials. Methods Appl. Anal. 6 (2), pp. 131–146.

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External Links:

ISSN 1073-2772, Pdf, MathReview, ZentralBlatt

Referenced by:

§18.15(vi), §18.16(vi).

Encodings:

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30: Errata

… ►The specific updates to Chapter 18 include some results for general orthogonal polynomials including quadratic transformations, uniqueness of orthogonality measure and completeness, moments, continued fractions, and some special classes of orthogonal polynomials. …We have significantly expanded the section on associated orthogonal polynomials, including expanded properties of associated Laguerre, Hermite, Meixner–Pollaczek, and corecursive orthogonal and numerator and denominator orthogonal polynomials. …We have also completely expanded our discussion on applications of orthogonal polynomials in the physical sciences, and also methods of computation for orthogonal polynomials. … ►

The following additions were made in Chapter 18:

• 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),

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

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

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

• 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.

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

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

• Equation (18.15.4_5).

• 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).

• 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).

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

• A new introduction.

• Equation (18.21.13).

• 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”.

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

• 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).

• 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).

• 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).

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

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

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

• 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).

• 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”.

• 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).

• 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 $$\left(\frac{-{\mathrm{\hslash }}^2}{2m}\frac{{\partial }^2}{{\partial x}^2}+V(x)\right)\psi (x,t)=i\mathrm{\hslash }\frac{\partial }{\partial t}\psi (x,t),$$

  where $\mathrm{\hslash }$ is the reduced Planck’s constant. On substituting $\psi (x,t)=\eta (x)\zeta (t)$, we obtain two ordinary differential equations, each of which involve the same constant $E$. The equation for $\eta (x)$ is

  errata.2 $$\frac{d^2\eta }{{dx}^2}+\frac{2m}{{\mathrm{\hslash }}^2}\left(E-V(x)\right)\eta =0.$$

  For a harmonic oscillator, the potential energy is given by

  errata.3 $$V(x)=\frac{1}{2}m{\omega }^2{x}^2,$$

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

  errata.4 $$E=E_n=\left(n+\frac{1}{2}\right)\mathrm{\hslash }\omega ,$$ $n=0,1,2,\mathrm{\dots }$.

  The corresponding eigenfunctions are

  errata.5 $${\eta }_n(x)={\pi }^{-\frac{1}{4}}{2}^{-\frac{1}{2}n}{(n!b)}^{-\frac{1}{2}}{H}_n\left(x/b\right){\mathrm{e}}^{-x^2/2b^2},$$

  where $b={(\mathrm{\hslash }/m\omega )}^{1/2}$, 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 $${\nabla }^2\psi +\frac{2m}{{\mathrm{\hslash }}^2}\left(E-V(𝐱)\right)\psi =0,$$

  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).”

• 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.

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The reference Ismail (2005) has been replaced throughout by the further corrected paperback version Ismail (2009).

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