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

§18.37 Classical OP’s in Two or More Variables

Keywords:: classical orthogonal polynomials
Permalink:: http://dlmf.nist.gov/18.37

§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

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

¶ Definition in Terms of Jacobi Polynomials

18.37.1 $\mathop{R^{{(\alpha)}}_{{m,n}}\/}\nolimits\!\left(re^{{i\theta}}\right)=e^{{i(% m-n)\theta}}r^{{|m-n|}}\frac{\mathop{P^{{(\alpha,|m-n|)}}_{{\min(m,n)}}\/}% \nolimits\!\left(2r^{2}-1\right)}{\mathop{P^{{(\alpha,|m-n|)}}_{{\min(m,n)}}\/% }\nolimits\!\left(1\right)},$ $r\geq 0$ , $\theta\in\Real$ , $\alpha>-1$ .

Defines:: $\mathop{R^{{(\alpha)}}_{{m,n}}\/}\nolimits\!\left(z\right)$ : disk polynomial
Symbols:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $\in$ : element of, : base of exponential function, : open interval, $\Real$ : real line (excluding infinity), : complex variable, : nonnegative integer and : nonnegative integer
Referenced by:: ¶ ‣ §18.37(ii)
Permalink:: http://dlmf.nist.gov/18.37.E1
Encodings:: TeX, pMML, png

¶ Orthogonality

18.37.2 $\iint\limits_{{x^{2}+y^{2}<1}}\mathop{R^{{(\alpha)}}_{{m,n}}\/}\nolimits\!% \left(x+iy\right)\mathop{R^{{(\alpha)}}_{{j,\ell}}\/}\nolimits\!\left(x-iy% \right)\*(1-x^{2}-y^{2})^{\alpha}dxdy=0,$ $m\neq j$ and/or $n\neq\ell$ .

Symbols:: : differential of , $\mathop{R^{{(\alpha)}}_{{m,n}}\/}\nolimits\!\left(z\right)$ : disk polynomial, : real variable, $\ell$ : nonnegative integer, : nonnegative integer, : nonnegative integer and : real variable
Permalink:: http://dlmf.nist.gov/18.37.E2
Encodings:: TeX, pMML, png

¶ Equivalent Definition

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

18.37.3 $\mathop{R^{{(\alpha)}}_{{m,n}}\/}\nolimits\!\left(z\right)=\sum_{{j=0}}^{{\min% (m,n)}}c_{j}z^{{m-j}}\conj{z}^{{n-j}},$

Symbols:: $\conj{z}$ : complex conjugate, $\mathop{R^{{(\alpha)}}_{{m,n}}\/}\nolimits\!\left(z\right)$ : disk polynomial, : open interval, : complex variable, : nonnegative integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/18.37.E3
Encodings:: TeX, pMML, png

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

18.37.4 $\iint\limits_{{x^{2}+y^{2}<1}}\mathop{R^{{(\alpha)}}_{{m,n}}\/}\nolimits\!% \left(x+iy\right)(x-iy)^{{m-j}}(x+iy)^{{n-j}}\*(1-x^{2}-y^{2})^{\alpha}dxdy=0,$ $j=1,2,\dots,\min(m,n)$ ;

Symbols:: : differential of , $\mathop{R^{{(\alpha)}}_{{m,n}}\/}\nolimits\!\left(z\right)$ : disk polynomial, : open interval, : real variable, : nonnegative integer, : nonnegative integer and : real variable
Permalink:: http://dlmf.nist.gov/18.37.E4
Encodings:: TeX, pMML, png

18.37.5 $\mathop{R^{{(\alpha)}}_{{m,n}}\/}\nolimits\!\left(1\right)=1.$

Symbols:: $\mathop{R^{{(\alpha)}}_{{m,n}}\/}\nolimits\!\left(z\right)$ : disk polynomial, : nonnegative integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/18.37.E5
Encodings:: TeX, pMML, png

¶ Explicit Representation

18.37.6 $\mathop{R^{{(\alpha)}}_{{m,n}}\/}\nolimits\!\left(z\right)=\sum_{{j=0}}^{{\min% (m,n)}}\frac{(-1)^{j}\left(\alpha+1\right)_{{m+n-j}}\left(-m\right)_{{j}}\left% (-n\right)_{{j}}}{\left(\alpha+1\right)_{{m}}\left(\alpha+1\right)_{{n}}j!}\*z% ^{{m-j}}\*\conj{z}^{{n-j}}.$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\conj{z}$ : complex conjugate, $\mathop{R^{{(\alpha)}}_{{m,n}}\/}\nolimits\!\left(z\right)$ : disk polynomial, : : factorial, : open interval, : complex variable, : nonnegative integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/18.37.E6
Encodings:: TeX, pMML, png

§18.37(ii) OP’s on the Triangle

Notes:: See Koornwinder (1975c).
Keywords:: orthogonal polynomials on the triangle
Permalink:: http://dlmf.nist.gov/18.37.ii

¶ Definition in Terms of Jacobi Polynomials

Keywords:: Jacobi polynomials, orthogonal polynomials on the triangle

18.37.7 $\mathop{P^{{\alpha,\beta,\gamma}}_{{m,n}}\/}\nolimits\!\left(x,y\right)=% \mathop{P^{{(\alpha,\beta+\gamma+2n+1)}}_{{m-n}}\/}\nolimits\!\left(2x-1\right% )\*x^{n}\mathop{P^{{(\beta,\gamma)}}_{{n}}\/}\nolimits\!\left(2x^{{-1}}y-1% \right),$ $m\geq n\geq 0$ , $\alpha,\beta,\gamma>-1$ .

Defines:: $\mathop{P^{{\alpha,\beta,\gamma}}_{{m,n}}\/}\nolimits\!\left(x,y\right)$ : triangle polynomial
Symbols:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, : real variable, : nonnegative integer, : nonnegative integer and : real variable
Referenced by:: ¶ ‣ §18.37(ii)
Permalink:: http://dlmf.nist.gov/18.37.E7
Encodings:: TeX, pMML, png

¶ Orthogonality

18.37.8 $\iint\limits_{{0<y<x<1}}\mathop{P^{{\alpha,\beta,\gamma}}_{{m,n}}\/}\nolimits% \!\left(x,y\right)\mathop{P^{{\alpha,\beta,\gamma}}_{{j,\ell}}\/}\nolimits\!% \left(x,y\right)\*(1-x)^{\alpha}(x-y)^{\beta}y^{\gamma}dxdy=0,$ $m\neq j$ and/or $n\neq\ell$ .

Symbols:: : differential of , $\mathop{P^{{\alpha,\beta,\gamma}}_{{m,n}}\/}\nolimits\!\left(x,y\right)$ : triangle polynomial, : real variable, $\ell$ : nonnegative integer, : nonnegative integer, : nonnegative integer and : real variable
Permalink:: http://dlmf.nist.gov/18.37.E8
Encodings:: TeX, pMML, png

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

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

Keywords:: orthogonal polynomials associated with root systems
Permalink:: http://dlmf.nist.gov/18.37.iii

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 -ultraspherical, or Askey–Wilson polynomials. In several variables they occur, for , 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 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).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 18.36 Miscellaneous Polynomials 18.38 Mathematical Applications

§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