orthogonal polynomials on the triangle

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

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§18.37(ii) OP’s on the Triangle

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Definition in Terms of Jacobi Polynomials

►

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:: pMML, png, TeX
See also:: Annotations for §18.37(ii), §18.37(ii), §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: 18.1 Notation

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

…

3: 37.3 Triangular Region with Weight Function $x^{\alpha}y^{\beta}(1-x-y)^{\gamma}$

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§37.3(ii) Orthogonal Bases

… ►They form an orthogonal basis of

\mathcal{V}_{n}^{\alpha,\beta,\gamma}

: …

4: 37.15 Orthogonal Polynomials on the Ball

… ►In particular, the various explicit bases of orthogonal or biorthogonal polynomials on

\triangle^{d}

are related to similar explicit bases on

\mathbb{B}^{d}

by quadratic transformations. …

5: 37.4 Disk with Weight Function $\left(1-x^{2}-y^{2}\right)^{\alpha}$

… ►In particular, the various explicit bases of orthogonal or biorthogonal polynomials on

\triangle

are related to similar explicit bases on

\mathbb{D}

by quadratic transformations. …

6: 37.14 Orthogonal Polynomials on the Simplex

… ►For

\alpha_{1},\ldots,\alpha_{d+1}>-1

the polynomials

P_{\boldsymbol{{\nu}}}^{\boldsymbol{{\alpha}}}

(

|{\boldsymbol{{\nu}}}|=n

) form an orthogonal basis of

\mathcal{V}_{n}^{\boldsymbol{{\alpha}}}(\triangle^{d})

. …

7: 37.2 General Orthogonal Polynomials of Two Variables

§37.2 General Orthogonal Polynomials of Two Variables

… ►the polynomials

P_{k,n}(x,y)

form an orthogonal system: … ►There are orthogonality relations … ►such that for some (not necessarily nonnegative) weight function the corresponding spaces

\mathcal{V}_{n}

of orthogonal polynomials are eigenspaces of

L

: … ►, having discrete orthogonal polynomials of two variables of degree

n

as eigenspaces. …

8: 37.16 Orthogonal Polynomials on the Hyperoctant

§37.16 Orthogonal Polynomials on the Hyperoctant

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§37.16(i) Orthogonal Bases

►Obviously, an orthogonal basis of

\mathcal{V}_{n}^{\boldsymbol{{\alpha}}}(\mathbb{R}_{+}^{d})

consisting of products of Laguerre polynomials is given by … ►where the Jacobi polynomials

P_{\boldsymbol{{\nu}}}^{\boldsymbol{{\alpha}}}

on the simplex

\triangle^{d-1}

are defined in §37.14(ii). … …

9: 37.10 Other Orthogonal Polynomials of Two Variables

§37.10 Other Orthogonal Polynomials of Two Variables

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§37.10(ii) Orthogonal Polynomials on an Annulus

… ► … ►There is also a limit to Jacobi polynomials on the triangle (see (37.3.3)): … ►Connection formulas between two such bases use

{{}_{4}F_{3}}

hypergeometric functions that can be written in terms of Racah polynomials, precisely as in §37.3(ii) for the Jacobi polynomials on the triangle. …

10: 37.7 Parabolic Biangular Region with Weight Function $\left(1-x\right)^{\alpha}\left(x-y^{2}\right)^{\beta}$

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§37.7(i) Jacobi polynomials on the parabolic biangular region $\mathbb{A}$

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§37.7(ii) Quadratic Transformations

►The Jacobi polynomials (37.3.3) on

\triangle

and the Jacobi polynomials (37.7.3) on

\mathbb{A}

are related by the quadratic transformations … ► … ► …