# central differences in imaginary direction

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##### 1: 18.1 Notation

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###### $x$-Differences

►Forward differences: … ►Backward differences: … ►Central differences in imaginary direction: … ►In Koekoek et al. (2010) ${\delta}_{x}$ denotes the operator $\mathrm{i}{\delta}_{x}$.##### 2: 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. …##### 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: 18.22 Hahn Class: Recurrence Relations and Differences

###### §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 $x$

… ►###### §18.22(iii) $x$-Differences

…##### 5: 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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##### 6: 18.26 Wilson Class: Continued

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###### §18.26(iii) Difference Relations

►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). ►For each family only the $y$-difference that lowers $n$ is given. … ►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
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###### §18.2(ii) $x$-Difference Operators

►If the orthogonality discrete set $X$ is $\{0,1,\mathrm{\dots},N\}$ or $\{0,1,2,\mathrm{\dots}\}$, then the role of the differentiation operator $d/dx$ in the case of classical OP’s (§18.3) is played by ${\mathrm{\Delta}}_{x}$, the forward-difference operator, or by ${\nabla}_{x}$, the backward-difference operator; compare §18.1(i). … ►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)). …##### 8: 16.25 Methods of Computation

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►They are similar to those described for confluent hypergeometric functions, and hypergeometric functions in §§13.29 and 15.19.
There is, however, an added feature in the numerical solution of differential equations and difference equations (recurrence relations).
This occurs when the wanted solution is intermediate in asymptotic growth compared with other solutions.
In these cases integration, or recurrence, in either a forward or a backward direction is unstable.
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##### 9: 18.25 Wilson Class: Definitions

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►For the Wilson class OP’s ${p}_{n}(x)$ with $x=\lambda (y)$: if the $y$-orthogonality set is $\{0,1,\mathrm{\dots},N\}$, then the role of the differentiation operator $d/dx$
in the Jacobi, Laguerre, and Hermite cases is played by the operator ${\mathrm{\Delta}}_{y}$ followed by division by ${\mathrm{\Delta}}_{y}(\lambda (y))$, or by the operator ${\nabla}_{y}$ followed by division by ${\nabla}_{y}(\lambda (y))$.
Alternatively if the $y$-orthogonality interval is $(0,\mathrm{\infty})$, then the role of $d/dx$ is played by the operator ${\delta}_{y}$ followed by division by ${\delta}_{y}(\lambda (y))$.
►Table 18.25.1 lists the transformations of variable, orthogonality ranges, and parameter constraints that are needed in §18.2(i) for the Wilson polynomials ${W}_{n}(x;a,b,c,d)$, continuous dual Hahn polynomials ${S}_{n}(x;a,b,c)$, Racah polynomials ${R}_{n}(x;\alpha ,\beta ,\gamma ,\delta )$, and dual Hahn polynomials ${R}_{n}(x;\gamma ,\delta ,N)$.
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18.25.4
$$w({y}^{2})=\frac{1}{2y}{\left|\frac{{\prod}_{j}\mathrm{\Gamma}\left({a}_{j}+\mathrm{i}y\right)}{\mathrm{\Gamma}\left(2\mathrm{i}y\right)}\right|}^{2},$$

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18.25.7
$$w({y}^{2})=\frac{1}{2y}{\left|\frac{{\prod}_{j}\mathrm{\Gamma}\left({a}_{j}+\mathrm{i}y\right)}{\mathrm{\Gamma}\left(2\mathrm{i}y\right)}\right|}^{2},$$

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##### 10: Sidebar 9.SB1: Supernumerary Rainbows

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►The faint line below the main colored arc is a ‘supernumerary rainbow’, produced by the interference of different sun-rays traversing a raindrop and emerging in the same direction.
…Airy invented his function in 1838 precisely to describe this phenomenon more accurately than Young had done in 1800 when pointing out that supernumerary rainbows require the wave theory of light and are impossible to explain with Newton’s picture of light as a stream of independent corpuscles.
The house in the picture is Newton’s birthplace.
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