# central in imaginary direction

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##### 1: Mourad E. H. Ismail

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► 1944, in Cairo, Egypt) is a Distinguished Research Professor in the Department of Mathematics of the University of Central Florida.
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►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).
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► 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

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►Central differences in imaginary direction:
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►The main functions treated in this chapter are:
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###### Classical OP’s in Two Variables

… ►In Szegő (1975, §4.7) the ultraspherical polynomials ${C}_{n}^{(\lambda )}\left(x\right)$ are denoted by ${P}_{n}^{(\lambda )}(x)$. … ►In Koekoek et al. (2010) ${\delta}_{x}$ denotes the operator $\mathrm{i}{\delta}_{x}$.##### 3: Gergő Nemes

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► 1988 in Szeged, Hungary) is a Research Fellow at the Alfréd Rényi Institute of Mathematics in Budapest, Hungary.
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► 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.
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##### 4: 18.20 Hahn Class: Explicit Representations

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►For comments on the use of the forward-difference operator ${\mathrm{\Delta}}_{x}$, the backward-difference operator ${\nabla}_{x}$, and the central-difference operator ${\delta}_{x}$, see §18.2(ii).
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►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.
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18.20.3
$$w(x;a,b,\overline{a},\overline{b}){p}_{n}(x;a,b,\overline{a},\overline{b})=\frac{1}{n!}{\delta}_{x}^{n}\left(w(x;a+\frac{1}{2}n,b+\frac{1}{2}n,\overline{a}+\frac{1}{2}n,\overline{b}+\frac{1}{2}n)\right).$$

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18.20.4
$${w}^{(\lambda )}(x;\varphi ){P}_{n}^{(\lambda )}(x;\varphi )=\frac{1}{n!}{\delta}_{x}^{n}\left({w}^{(\lambda +\frac{1}{2}n)}(x;\varphi )\right).$$

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##### 5: 18.22 Hahn Class: Recurrence Relations and Differences

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###### §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
$${\delta}_{x}\left({p}_{n}(x;a,b,\overline{a},\overline{b})\right)=(n+2\mathrm{\Re}\left(a+b\right)-1){p}_{n-1}(x;a+\frac{1}{2},b+\frac{1}{2},\overline{a}+\frac{1}{2},\overline{b}+\frac{1}{2}),$$

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##### 6: 18.26 Wilson Class: Continued

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►For comments on the use of the forward-difference operator ${\mathrm{\Delta}}_{x}$, the backward-difference operator ${\nabla}_{x}$, and the central-difference operator ${\delta}_{x}$, see §18.2(ii).
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18.26.14
$${\delta}_{y}\left({W}_{n}({y}^{2};a,b,c,d)\right)/{\delta}_{y}({y}^{2})=-n(n+a+b+c+d-1){W}_{n-1}({y}^{2};a+\frac{1}{2},b+\frac{1}{2},c+\frac{1}{2},d+\frac{1}{2}).$$

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18.26.15
$${\delta}_{y}\left({S}_{n}({y}^{2};a,b,c)\right)/{\delta}_{y}({y}^{2})=-n{S}_{n-1}({y}^{2};a+\frac{1}{2},b+\frac{1}{2},c+\frac{1}{2}).$$

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►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.
##### 7: 18.2 General Orthogonal Polynomials

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►Let $(a,b)$ be a finite or infinite open interval in
$\mathbb{R}$.
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►In the former case we also require
…whereas in the latter case the system $\{{p}_{n}(x)\}$ is finite: $n=0,1,\mathrm{\dots},N$.
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►If the orthogonality interval is $(-\mathrm{\infty},\mathrm{\infty})$ or $(0,\mathrm{\infty})$, then the role of $d/dx$ can be played by ${\delta}_{x}$, the central-difference operator in the imaginary direction (§18.1(i)).
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►As in §18.1(i) we assume that ${p}_{-1}(x)\equiv 0$.
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##### 8: 10.73 Physical Applications

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►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.
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►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.
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►More recently, Bessel functions appear in the inverse problem in wave propagation, with applications in medicine, astronomy, and acoustic imaging.
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►In quantum mechanics the spherical Bessel functions arise in the solution of the Schrödinger wave equation for a particle in a central potential.
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##### 9: 36.15 Methods of Computation

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►Close to the origin $\mathbf{x}=\mathbf{0}$ of parameter space, the series in §36.8 can be used.
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►Far from the bifurcation set, the leading-order asymptotic formulas of §36.11 reproduce accurately the form of the function, including the geometry of the zeros described in §36.7.
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►Direct numerical evaluation can be carried out along a contour that runs along the segment of the real $t$-axis containing all real critical points of $\mathrm{\Phi}$ and is deformed outside this range so as to reach infinity along the asymptotic valleys of $\mathrm{exp}\left(\mathrm{i}\mathrm{\Phi}\right)$.
…There is considerable freedom in the choice of deformations.
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►This can be carried out by direct numerical evaluation of canonical integrals along a finite segment of the real axis including all real critical points of $\mathrm{\Phi}$, with contributions from the contour outside this range approximated by the first terms of an asymptotic series associated with the endpoints.
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##### 10: Staff

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►Frank W. J. Olver [December 15, 1924-April 23, 2013] served as Editor-in-Chief and Mathematics Editor for the DLMF project from its inception until his death on April 23, 2013.
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Ian J. Thompson, Lawrence Livermore National Laboratory, *Chap.* 33

Mourad E. H. Ismail, University of Central Florida

Ian J. Thompson, Lawrence Livermore National Laboratory,
*for Chap.* 33