central in imaginary direction

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1: Mourad E. H. Ismail

… … ► 1944, in Cairo, Egypt) is a Distinguished Research Professor in the Department of Mathematics of the University of Central Florida. … ►His well-known book Classical and Quantum Orthogonal Polynomials in One Variable was published by Cambridge University Press in 2005 and reprinted with corrections in paperback in Ismail (2009). … ► 254, American Mathematical Society, 2000; Special Functions—Proceedings of the International Workshop, Hong Kong, June 21–25, 1999, World Scientific, 2000; Special Functions 2000: Current Perspective and Future Directions (with J. … Koelink), Developments in Mathematics, v. …

2: 18.1 Notation

… ►Central differences in imaginary direction: … ►The main functions treated in this chapter are: … ►

Associated OP’s are denoted via addition of the letter $c$ at the end of the listing of parameters in their usual notations.

… ►

Classical OP’s in Two Variables

… ►In Koekoek et al. (2010)

\delta_{x}

denotes the operator

\mathrm{i}\delta_{x}

3: Gergő Nemes

… ► 1988 in Szeged, Hungary) is a Professor in the School of Mathematics at the Harbin Institute of Technology, China. … ► in mathematics (with distinction) and a M. …in mathematics (with honours) from Loránd Eötvös University, Budapest, Hungary and a Ph. … in mathematics from Central European University in Budapest, Hungary. ►Nemes has research interests in asymptotic analysis, Écalle theory, exact WKB analysis, and special functions. …

4: 18.40 Methods of Computation

… ► … ►A simple set of choices is spelled out in Gordon (1968) which gives a numerically stable algorithm for direct computation of the recursion coefficients in terms of the moments, followed by construction of the J-matrix and quadrature weights and abscissas, and we will follow this approach: Let

N

be a positive integer and define … ►in which …The question is then: how is this possible given only

F_{N}(z)

, rather than

F(z)

itself?

F_{N}(z)

often converges to smooth results for

z

off the real axis for

\Im{z}

at a distance greater than the pole spacing of the

x_{n}

, this may then be followed by approximate numerical analytic continuation via fitting to lower order continued fractions (either Padé, see §3.11(iv), or pointwise continued fraction approximants, see Schlessinger (1968, Appendix)), to

F_{N}(z)

and evaluating these on the real axis in regions of higher pole density that those of the approximating function. … ►The quadrature points and weights can be put to a more direct and efficient use. …

5: 37.21 Physical Applications

… ►When, for an optical system, the aberration function of a circular wavefront is expanded in terms of these polynomials, terms with large coefficients give insight in the type of aberration, see Born and Wolf (1999, Ch. IX). …This method is used in many specialized fields, among which ophthalmology, see McAlinden et al. (2011)). A very important application is in lithography (see references in de Winter et al. (2020)), where Zernike polynomials are still used in the recent EUV (extreme ultraviolet) lithography developed by ASML and Zeiss companies. In a next step in EUV technology the NA (numerical aperture) will be further increased, which goes with a central pupil obscuration. … …

6: 18.20 Hahn Class: Explicit Representations

… ►For comments on the use of the forward-difference operator

\Delta_{x}

, the backward-difference operator

\nabla_{x}

, and the central-difference operator

\delta_{x}

, see §18.2(ii). … ►In (18.20.1)

X

and

w_{x}

are as in Table 18.19.1. …For the Krawtchouk, Meixner, and Charlier polynomials,

F(x)

and

\kappa_{n}

are as in Table 18.20.1. … ►

18.20.3

w(x;a,b,\overline{a},\overline{b})p_{n}\left(x;a,b,\overline{a},\overline{b}% \right)=\frac{1}{n!}\delta_{x}^{n}\left(w(x;a+\tfrac{1}{2}n,b+\tfrac{1}{2}n,% \overline{a}+\tfrac{1}{2}n,\overline{b}+\tfrac{1}{2}n)\right).

ⓘ

Symbols:: $\delta_{\NVar{x}}$ : central difference, $\overline{\NVar{z}}$ : complex conjugate, $p_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{\overline{a}},\NVar{% \overline{b}}\right)$ : continuous Hahn polynomial, $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.20(i)
Permalink:: http://dlmf.nist.gov/18.20.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §18.20(i), §18.20(i), §18.20 and Ch.18

… ►

18.20.4

w^{(\lambda)}(x;\phi)P^{(\lambda)}_{n}\left(x;\phi\right)=\frac{1}{n!}\delta_{% x}^{n}\left(w^{(\lambda+\frac{1}{2}n)}(x;\phi)\right).

ⓘ

Symbols:: $P^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x};\NVar{\phi}\right)$ : Meixner–Pollaczek polynomial, $\delta_{\NVar{x}}$ : central difference, $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.20(i)
Permalink:: http://dlmf.nist.gov/18.20.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §18.20(i), §18.20(i), §18.20 and Ch.18

…

7: 18.22 Hahn Class: Recurrence Relations and Differences

… ►

§18.22(i) Recurrence Relations in $n$

… ►These polynomials satisfy (18.22.2) with

p_{n}(x)

A_{n}

, and

C_{n}

as in Table 18.22.1. … ►

§18.22(ii) Difference Equations in $x$

… ►For

A(x)

C(x)

, and

\lambda_{n}

in (18.22.12) see Table 18.22.2. … ►

18.22.27

\delta_{x}\left(p_{n}\left(x;a,b,\overline{a},\overline{b}\right)\right)=(n+2% \Re\left(a+b\right)-1)\*p_{n-1}\left(x;a+\tfrac{1}{2},b+\tfrac{1}{2},\overline% {a}+\tfrac{1}{2},\overline{b}+\tfrac{1}{2}\right),

ⓘ

Symbols:: $\delta_{\NVar{x}}$ : central difference, $\overline{\NVar{z}}$ : complex conjugate, $p_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{\overline{a}},\NVar{% \overline{b}}\right)$ : continuous Hahn polynomial, $\Re$ : real part, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.22(iii)
Permalink:: http://dlmf.nist.gov/18.22.E27
Encodings:: pMML, png, TeX
See also:: Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18

…

8: 18.26 Wilson Class: Continued

… ►For comments on the use of the forward-difference operator

\Delta_{x}

, the backward-difference operator

\nabla_{x}

, and the central-difference operator

\delta_{x}

, see §18.2(ii). … ►

18.26.14

\ifrac{\delta_{y}\left(W_{n}\left(y^{2};a,b,c,d\right)\right)}{\delta_{y}(y^{2% })}=-n(n+a+b+c+d-1)\*W_{n-1}\left(y^{2};a+\tfrac{1}{2},b+\tfrac{1}{2},c+\tfrac% {1}{2},d+\tfrac{1}{2}\right).

ⓘ

Symbols:: $W_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c},\NVar{d}\right)$ : Wilson polynomial, $\delta_{\NVar{x}}$ : central difference, $y$ : real variable and $n$ : nonnegative integer
Referenced by:: §16.4(iii), §18.26(iii)
Permalink:: http://dlmf.nist.gov/18.26.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §18.26(iii), §18.26 and Ch.18

►

18.26.15

\ifrac{\delta_{y}\left(S_{n}\left(y^{2};a,b,c\right)\right)}{\delta_{y}(y^{2})% }=-nS_{n-1}\left(y^{2};a+\tfrac{1}{2},b+\tfrac{1}{2},c+\tfrac{1}{2}\right).

ⓘ

Symbols:: $\delta_{\NVar{x}}$ : central difference, $S_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c}\right)$ : continuous dual Hahn polynomial, $y$ : real variable and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/18.26.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §18.26(iii), §18.26 and Ch.18

… ►Koornwinder (2009) rescales and reparametrizes Racah polynomials and Wilson polynomials in such a way that they are continuous in their four parameters, provided that these parameters are nonnegative. Moreover, if one or more of the new parameters becomes zero, then the polynomial descends to a lower family in the Askey scheme.

9: 37.20 Mathematical Applications

… ►Partial sums of Fourier orthogonal polynomial expansions are polynomials of best approximation in

L^{2}\left(W\right)

space and they are also the essential building blocks for approximation in

L^{p}

spaces. … ►Complex circular Hermite polynomials (see §37.6(i)) are also used in the physics models, see the references in (Ismail, 2016, §1). … ►OPs are used in collocation method or spectral method for numerical solution of partial differential equations. In the latter method, the approximating functions are taken as a linear combinations of OPs and their coefficients are determined by the Galerkin method. … ►Although Gaussian cubature rules rarely exist and they do not exist for centrally symmetric domains, minimal or near minimal cubature rules on the unit square are known and provide efficient numerical integration rules. …

10: 10.73 Physical Applications

… ►Laplace’s equation governs problems in heat conduction, in the distribution of potential in an electrostatic field, and in hydrodynamics in the irrotational motion of an incompressible fluid. … ►This equation governs problems in acoustic and electromagnetic wave propagation. …Consequently, Bessel functions

J_{n}\left(x\right)

, and modified Bessel functions

I_{n}\left(x\right)

, are central to the analysis of microwave and optical transmission in waveguides, including coaxial and fiber. … ►More recently, Bessel functions appear in the inverse problem in wave propagation, with applications in medicine, astronomy, and acoustic imaging. … ►In quantum mechanics the spherical Bessel functions arise in the solution of the Schrödinger wave equation for a particle in a central potential. …