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

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18.37.2 $m\ne j$ and/or $n\ne \mathrm{\ell }$.

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Symbols:

$dx$: differential of $x$ , $R_{m,n}^{(\alpha )}\left(z\right)$: disk polynomial, $\mathrm{i}$: imaginary unit, $y$: real variable, $\mathrm{\ell }$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.37.E2

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Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

… ►The following three conditions, taken together, determine $R_{m,n}^{(\alpha )}\left(z\right)$ uniquely: … ►

18.37.4 $j=1,2,\mathrm{\dots },\min (m,n)$;

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Symbols:

$dx$: differential of $x$ , $R_{m,n}^{(\alpha )}\left(z\right)$: 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:

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Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

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18.37.5 $$R_{m,n}^{(\alpha )}\left(1\right)=1.$$

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Symbols:

$R_{m,n}^{(\alpha )}\left(z\right)$: 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 $m\ne j$ and/or $n\ne \mathrm{\ell }$.

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Symbols:

$dx$: differential of $x$ , $P_{m,n}^{\alpha ,\beta ,\gamma }(x,y)$: triangle polynomial, $y$: real variable, $\mathrm{\ell }$: 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: 37 Orthogonal Polynomials of Several Variables

Chapter 37 Orthogonal Polynomials of Several Variables

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3: 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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4: 18 Orthogonal Polynomials

Chapter 18 Orthogonal Polynomials

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5: Yuan Xu

… ►Xu has published numerous papers on analysis including approximation theory, harmonic analysis, orthogonal polynomials, numerical analysis, and special functions. His interest is mostly on higher dimensional problems, such as orthogonal polynomials of several variables, cubature formulas, and mutivariable approximation. ►His well-known book Orthogonal Polynomials of Several Variables (with C. … ►Xu has been active as an officer in the SIAM Activity Group on Orthogonal Polynomials and Special Functions; in 2014-2015 and in 2017-2019 he served as Secretary. … ►

37 Orthogonal Polynomials of Several Variables

6: 16.7 Relations to Other Functions

§16.7 Relations to Other Functions

►For orthogonal polynomials see Chapter 18. …

7: 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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8: 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: $R_{m,n}^{(\alpha )}\left(z\right)$.

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Triangle: $P_{m,n}^{\alpha ,\beta ,\gamma }(x,y)$.

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9: 32.15 Orthogonal Polynomials

§32.15 Orthogonal Polynomials

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10: 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. …