# §18.37 Classical OP’s in Two or More Variables

## §18.37(i) Disk Polynomials

### Definition in Terms of Jacobi Polynomials

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

 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 $n\neq\ell$.

### Equivalent Definition

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

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

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

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

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

## §18.37(ii) OP’s on the Triangle

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

### Orthogonality

 18.37.8 $\iint\limits_{0 $m\neq j$ and/or $n\neq\ell$.

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

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).