18 Orthogonal Polynomials Other Orthogonal Polynomials 18.36 Miscellaneous Polynomials 18.38 Mathematical Applications

§18.37 Classical OP’s in Two or More Variables

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Keywords:: classical orthogonal polynomials, in two or more variables
Permalink:: http://dlmf.nist.gov/18.37
See also:: Annotations for Ch.18

§18.37(i) Disk Polynomials
§18.37(ii) OP’s on the Triangle
§18.37(iii) OP’s Associated with Root Systems

§18.37(i) Disk Polynomials

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Keywords:: disk polynomials
Notes:: See Dunkl and Xu (2001, §2.4.3).
Permalink:: http://dlmf.nist.gov/18.37.i
See also:: Annotations for §18.37 and Ch.18

Definition in Terms of Jacobi Polynomials

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

18.37.1		$R^{(\alpha)}_{m,n}\left(re^{\mathrm{i}\theta}\right)=e^{\mathrm{i}(m-n)\theta}% r^{\|m-n\|}\frac{P^{(\alpha,\|m-n\|)}_{\min(m,n)}\left(2r^{2}-1\right)}{P^{(\alpha% ,\|m-n\|)}_{\min(m,n)}\left(1\right)},$
$r\geq 0$ , $\theta\in\mathbb{R}$ , $\alpha>-1$ .
ⓘ Defines: $R^{(\NVar{\alpha})}_{\NVar{m},\NVar{n}}\left(\NVar{z}\right)$ : disk polynomial Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $\in$ : element of, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\mathbb{R}$ : real line, $z$ : complex variable, $m$ : nonnegative integer and $n$ : nonnegative integer Referenced by: §18.37(ii) Permalink: http://dlmf.nist.gov/18.37.E1 Encodings: TeX, pMML, png See also: Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

Orthogonality

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

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: TeX, pMML, png See also: Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

Equivalent Definition

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

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

18.37.3		$R^{(\alpha)}_{m,n}\left(z\right)=\sum_{j=0}^{\min(m,n)}c_{j}z^{m-j}{\overline{% z}}^{n-j},$
ⓘ Symbols: $\overline{\NVar{z}}$ : complex conjugate, $R^{(\NVar{\alpha})}_{\NVar{m},\NVar{n}}\left(\NVar{z}\right)$ : disk polynomial, $z$ : complex variable, $m$ : nonnegative integer and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/18.37.E3 Encodings: TeX, pMML, png See also: Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

where $c_{j}$ are real or complex constants, with $c_{0}\neq 0$ ;

18.37.4		$\iint\limits_{x^{2}+y^{2}<1}R^{(\alpha)}_{m,n}\left(x+\mathrm{i}y\right)(x-iy)% ^{m-j}(x+iy)^{n-j}\*(1-x^{2}-y^{2})^{\alpha}\mathrm{d}x\mathrm{d}y=0,$
$j=1,2,\dots,\min(m,n)$ ;
ⓘ 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: TeX, pMML, png See also: Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

18.37.5		$R^{(\alpha)}_{m,n}\left(1\right)=1.$
ⓘ 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: TeX, pMML, png See also: Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

Explicit Representation

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

18.37.6		$R^{(\alpha)}_{m,n}\left(z\right)=\sum_{j=0}^{\min(m,n)}\frac{(-1)^{j}{\left(% \alpha+1\right)_{m+n-j}}{\left(-m\right)_{j}}{\left(-n\right)_{j}}}{{\left(% \alpha+1\right)_{m}}{\left(\alpha+1\right)_{n}}j!}\z^{m-j}\{\overline{z}}^{n% -j}.$
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\overline{\NVar{z}}$ : complex conjugate, $R^{(\NVar{\alpha})}_{\NVar{m},\NVar{n}}\left(\NVar{z}\right)$ : disk polynomial, $!$ : factorial (as in $n!$ ), $z$ : complex variable, $m$ : nonnegative integer and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/18.37.E6 Encodings: TeX, pMML, png See also: Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

§18.37(ii) OP’s on the Triangle

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Keywords:: orthogonal polynomials on the triangle
Notes:: See Koornwinder (1975c).
Permalink:: http://dlmf.nist.gov/18.37.ii
See also:: Annotations for §18.37 and Ch.18

Definition in Terms of Jacobi Polynomials

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Keywords:: Jacobi polynomials, orthogonal polynomials on the triangle, relations to other functions
See also:: Annotations for §18.37(ii), §18.37 and Ch.18

18.37.7		$P^{\alpha,\beta,\gamma}_{m,n}\left(x,y\right)=P^{(\alpha,\beta+\gamma+2n+1)}_{% m-n}\left(2x-1\right)\*x^{n}P^{(\beta,\gamma)}_{n}\left(2x^{-1}y-1\right),$
$m\geq n\geq 0$ , $\alpha,\beta,\gamma>-1$ .
ⓘ Defines: $P^{\NVar{\alpha},\NVar{\beta},\NVar{\gamma}}_{\NVar{m},\NVar{n}}\left(\NVar{x}% ,\NVar{y}\right)$ : triangle polynomial Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $y$ : real variable, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.37(ii) Permalink: http://dlmf.nist.gov/18.37.E7 Encodings: TeX, pMML, png See also: Annotations for §18.37(ii), §18.37(ii), §18.37 and Ch.18

Orthogonality

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See also:: Annotations for §18.37(ii), §18.37 and Ch.18

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: TeX, pMML, png See also: Annotations for §18.37(ii), §18.37(ii), §18.37 and Ch.18

See Dunkl and Xu (2001, §2.3.3) for analogs of (18.37.1) and (18.37.7) on a $d$ -dimensional simplex.

§18.37(iii) OP’s Associated with Root Systems

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Keywords:: orthogonal polynomials associated with root systems
Permalink:: http://dlmf.nist.gov/18.37.iii
See also:: Annotations for §18.37 and Ch.18

Orthogonal polynomials associated with root systems are certain systems of trigonometric polynomials in several variables, symmetric under a certain finite group (Weyl group), and orthogonal on a torus. In one variable they are essentially ultraspherical, Jacobi, continuous $q$ -ultraspherical, or Askey–Wilson polynomials. In several variables they occur, for $q=1$ , as Jack polynomials and also as Jacobi polynomials associated with root systems; see Macdonald (1995, Chapter VI, §10), Stanley (1989), Kuznetsov and Sahi (2006, Part 1), Heckman (1991). For general $q$ they occur as Macdonald polynomials for root system $A_{n}$ , as Macdonald polynomials for general root systems, and as Macdonald-Koornwinder polynomials; see Macdonald (1995, Chapter VI), Macdonald (2000, 2003), Koornwinder (1992).

§18.37 Classical OP’s in Two or More Variables

Contents

§18.37(i) Disk Polynomials

Definition in Terms of Jacobi Polynomials

Orthogonality

Equivalent Definition

Explicit Representation

§18.37(ii) OP’s on the Triangle

Definition in Terms of Jacobi Polynomials

Orthogonality

§18.37(iii) OP’s Associated with Root Systems