orthogonal polynomials

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1: 18.37 Classical OP’s in Two or More Variables

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18.37.2

\iint\limits_{x^{2}+y^{2}<1}R^{(\alpha)}_{m,n}\left(x+\mathrm{i}y\right)R^{(% \alpha)}_{j,\ell}\left(x-\mathrm{i}y\right)\*(1-x^{2}-y^{2})^{\alpha}\,\mathrm% {d}x\,\mathrm{d}y=0,

m\neq j

and/or

n\neq\ell

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $R^{(\NVar{\alpha})}_{\NVar{m},\NVar{n}}\left(\NVar{z}\right)$ : disk polynomial, $\mathrm{i}$ : imaginary unit, $y$ : real variable, $\ell$ : nonnegative integer, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.37.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

… ►The following three conditions, taken together, determine

R^{(\alpha)}_{m,n}\left(z\right)

uniquely: … ►

18.37.4

\iint\limits_{x^{2}+y^{2}<1}R^{(\alpha)}_{m,n}\left(x+\mathrm{i}y\right)(x-iy)% ^{m-j}(x+iy)^{n-j}\*(1-x^{2}-y^{2})^{\alpha}\,\mathrm{d}x\,\mathrm{d}y=0,

j=1,2,\dots,\min(m,n)

;

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $R^{(\NVar{\alpha})}_{\NVar{m},\NVar{n}}\left(\NVar{z}\right)$ : disk polynomial, $\mathrm{i}$ : imaginary unit, $y$ : real variable, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.37.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

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18.37.5

R^{(\alpha)}_{m,n}\left(1\right)=1.

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Symbols:: $R^{(\NVar{\alpha})}_{\NVar{m},\NVar{n}}\left(\NVar{z}\right)$ : disk polynomial, $m$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/18.37.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

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18.37.8

\iint\limits_{0<y<x<1}P^{\alpha,\beta,\gamma}_{m,n}\left(x,y\right)P^{\alpha,% \beta,\gamma}_{j,\ell}\left(x,y\right)\*(1-x)^{\alpha}(x-y)^{\beta}y^{\gamma}% \,\mathrm{d}x\,\mathrm{d}y=0,

m\neq j

and/or

n\neq\ell

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $P^{\NVar{\alpha},\NVar{\beta},\NVar{\gamma}}_{\NVar{m},\NVar{n}}\left(\NVar{x}% ,\NVar{y}\right)$ : triangle polynomial, $y$ : real variable, $\ell$ : nonnegative integer, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.37.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §18.37(ii), §18.37(ii), §18.37 and Ch.18

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2: René F. Swarttouw

… ►Swarttouw is mainly a teacher of mathematics and has published a few papers on special functions and orthogonal polynomials. He is coauthor of the book Hypergeometric Orthogonal Polynomials and Their $q$ -Analogues) Hypergeometric Orthogonal Polynomials and Their

q

-Analogues. … ►

18 Orthogonal Polynomials

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3: 18 Orthogonal Polynomials

Chapter 18 Orthogonal Polynomials

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4: 16.7 Relations to Other Functions

§16.7 Relations to Other Functions

►For orthogonal polynomials see Chapter 18. …

5: 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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6: 18.1 Notation

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$x, y, t$	real variables.
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OP’s	orthogonal polynomials.
EOP’s	exceptional orthogonal polynomials.

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Hahn Class OP’s

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Disk: $R^{(\alpha)}_{m,n}\left(z\right)$ .

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Triangle: $P^{\alpha,\beta,\gamma}_{m,n}\left(x,y\right)$ .

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7: 32.15 Orthogonal Polynomials

§32.15 Orthogonal Polynomials

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8: Wolter Groenevelt

… ►Groenevelt’s research interests is in special functions and orthogonal polynomials and their relations with representation theory and interacting particle systems. ►As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. …

9: 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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See accompanying text — Figure 18.21.1: Askey scheme. … Magnify

10: 18.3 Definitions

§18.3 Definitions

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With the property that $\{p_{n+1}^{\prime}(x)\}_{n=0}^{\infty}$ is again a system of OP’s. See §18.9(iii).

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As given by a Rodrigues formula (18.5.5).

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Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, standardizations, leading coefficients, and parameter constraints. …

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Name	$p_{n}(x)$	$(a,b)$	$w(x)$	$h_{n}$	$k_{n}$	$\ifrac{\tilde{k}_{n}}{k_{n}}$	Constraints
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► … ►For explicit power series coefficients up to

n=12

for these polynomials and for coefficients up to

n=6

for Jacobi and ultraspherical polynomials see Abramowitz and Stegun (1964, pp. 793–801). …