type 2 Pollaczek polynomials

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1: 18.35 Pollaczek Polynomials

… ► … ►we have the explicit representations … ► … … ►Then …

2: 18.30 Associated OP’s

… ►They can be expressed in terms of type 3 Pollaczek polynomials (which are also associated type 2 Pollaczek polynomials) by (18.35.10). … ►The type 3 Pollaczek polynomials are the associated type 2 Pollaczek polynomials, see §18.35. …

3: 18.39 Applications in the Physical Sciences

… ►The recursion of (18.39.46) is that for the type 2 Pollaczek polynomials of (18.35.2), with

\lambda=l+1

a=-b=2Z/s

, and

c=0

, and terminates for

x=x^{N}_{i}

being a zero of the polynomial of order

N

. …

4: Errata

… ►

Equation (18.35.1)

18.35.1

P^{{(\lambda)}}_{-1}\left(x;a,b,c\right)=0,

P^{{(\lambda)}}_{0}\left(x;a,b,c\right)=1

These equations which were previously given for Pollaczek polynomials of type 2 has been updated for Pollaczek polynomials of type 3.

►

Equation (18.35.2)

18.35.2

P^{{(\lambda)}}_{n+1}\left(x;a,b,c\right)=\frac{2(n+c+\lambda+a)x+2b}{n+c+1}\,% P^{{(\lambda)}}_{n}\left(x;a,b,c\right)-\frac{n+c+2\lambda-1}{n+c+1}\,P^{{(% \lambda)}}_{n-1}\left(x;a,b,c\right),

n=0,1,\dots

This recurrence relation which was previously given for Pollaczek polynomials of type 2 (the case $c=0$ ) has been updated for Pollaczek polynomials of type 3.

… ►

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

where $\hbar$ 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 $\frac{{\mathrm{d}}^{2}\eta}{{\mathrm{d}x}^{2}}+\frac{2m}{\hbar^{2}}\left(E-V(x% )\right)\eta=0.$

For a harmonic oscillator, the potential energy is given by

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

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

errata.4 $E=E_{n}=\left(n+\tfrac{1}{2}\right)\hbar\omega,$ $n=0,1,2,\dots$ .

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

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).”
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.

…

5: Bibliography K

… ►

T. Kasuga and R. Sakai (2003) Orthonormal polynomials with generalized Freud-type weights. J. Approx. Theory 121 (1), pp. 13–53.

ⓘ

External Links:: ISSN 0021-9045, Document, Link, MathReview (Walter Van Assche)
Referenced by:: §18.32.
Encodings:: BibTeX

… ►

J. Koekoek, R. Koekoek, and H. Bavinck (1998) On differential equations for Sobolev-type Laguerre polynomials. Trans. Amer. Math. Soc. 350 (1), pp. 347–393.

ⓘ

External Links:: ISSN 0002-9947, FullText, MathReview (Hans W. Volkmer), ZentralBlatt
Referenced by:: §18.36(ii).
Encodings:: BibTeX

… ►

T. H. Koornwinder (1989) Meixner-Pollaczek polynomials and the Heisenberg algebra. J. Math. Phys. 30 (4), pp. 767–769.

ⓘ

External Links:: ISSN 0022-2488, Document, MathReview, ZentralBlatt
Referenced by:: §18.21(ii).
Encodings:: BibTeX

►

T. H. Koornwinder (1992) Askey-Wilson Polynomials for Root Systems of Type

B C

. In Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991), Contemp. Math., Vol. 138, pp. 189–204.

ⓘ

External Links:: FullText, MathReview (Eric M. Opdam), ZentralBlatt
Referenced by:: §18.37(iii).
Encodings:: BibTeX

… ►

T. Koornwinder, A. Kostenko, and G. Teschl (2018) Jacobi polynomials, Bernstein-type inequalities and dispersion estimates for the discrete Laguerre operator. Adv. Math. 333, pp. 796–821.

ⓘ

External Links:: ISSN 0001-8708, Document, Link, MathReview (Martin Axel Hallnäs)
Referenced by:: §18.14(i).
Encodings:: BibTeX

…

6: Bibliography P

… ►

R. B. Paris (2005a) A Kummer-type transformation for a

{}_{2}F_{2}

hypergeometric function. J. Comput. Appl. Math. 173 (2), pp. 379–382.

ⓘ

External Links:: ISSN 0377-0427, Document, MathReview (P. Anandani), ZentralBlatt
Referenced by:: §16.6.
Encodings:: BibTeX

… ►

F. Pollaczek (1949a) Sur une généralisation des polynomes de Legendre. C. R. Acad. Sci. Paris 228, pp. 1363–1365.

ⓘ

External Links:: ISSN 0001-4036, MathReview (G. Szegö)
Referenced by:: §18.35(i).
Encodings:: BibTeX

►

F. Pollaczek (1949b) Systèmes de polynomes biorthogonaux qui généralisent les polynomes ultrasphériques. C. R. Acad. Sci. Paris 228, pp. 1998–2000.

ⓘ

External Links:: ISSN 0001-4036, MathReview (G. Szegö)
Referenced by:: §18.35(i).
Encodings:: BibTeX

►

F. Pollaczek (1950) Sur une famille de polynômes orthogonaux à quatre paramètres. C. R. Acad. Sci. Paris 230, pp. 2254–2256.

ⓘ

External Links:: MathReview (A. Erdélyi)
Referenced by:: (18.30.17), (18.30.18), (18.35.6_5), (18.35.6_6), §18.35(i).
Encodings:: BibTeX

… ►

S. Porubský (1998) Voronoi type congruences for Bernoulli numbers. In Voronoi’s Impact on Modern Science. Book I, P. Engel and H. Syta (Eds.),

ⓘ

External Links:: ZentralBlatt
Referenced by:: §24.10(iii).
Encodings:: BibTeX

…

7: Bibliography B

… ►

H. Bateman (1905) A generalisation of the Legendre polynomial. Proc. London Math. Soc. (2) 3 (3), pp. 111–123.

ⓘ

Notes:: Reviewed in JFM 36.0512.01
External Links:: Document, ZentralBlatt
Referenced by:: §18.12.
Encodings:: BibTeX

… ►

G. Baxter (1961) Polynomials defined by a difference system. J. Math. Anal. Appl. 2 (2), pp. 223–263.

ⓘ

External Links:: Document, MathReview (F. L. Spitzer), ZentralBlatt
Referenced by:: §18.33(v).
Encodings:: BibTeX

… ►

R. Bo and R. Wong (1996) Asymptotic behavior of the Pollaczek polynomials and their zeros. Stud. Appl. Math. 96, pp. 307–338.

ⓘ

External Links:: ISSN 0022-2526, MathReview (Harold Exton), ZentralBlatt
Referenced by:: §18.35(iii).
Encodings:: BibTeX

… ►

S. Bochner (1952) Bessel functions and modular relations of higher type and hyperbolic differential equations. Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.

ⓘ

External Links:: MathReview (S. C. van Veen), ZentralBlatt
Referenced by:: §35.5(i).
Encodings:: BibTeX

… ►

C. Brezinski (1980) Padé-type Approximation and General Orthogonal Polynomials. International Series of Numerical Mathematics, Vol. 50, Birkhäuser Verlag, Basel.

ⓘ

External Links:: ISBN 3-7643-1100-2, MathReview (A. Magnus), ZentralBlatt
Referenced by:: §3.11(iv).
Encodings:: BibTeX

…

8: 18.2 General Orthogonal Polynomials

… ►This happens, for example, with the continuous Hahn polynomials and Meixner–Pollaczek polynomials (§18.20(i)). … ►

Kernel Polynomials

… ►If these

x_{k}

satisfy

\sum_{k}(|x_{k}|-1)^{\ifrac{1}{2}}<\infty

then Szegő type asymptotics outside

[-1,1]

can be given for the corresponding OP’s, see Simon (2011, Corollary 3.7.2 and following). … ►

Sheffer Polynomials

… ►The generating functions (18.12.13), (18.12.15), (18.23.3), (18.23.4), (18.23.5) and (18.23.7) for Laguerre, Hermite, Krawtchouk, Meixner, Charlier and Meixner–Pollaczek polynomials, respectively, can be written in the form (18.2.45). …