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1: 37.2 General Orthogonal Polynomials of Two Variables

§37.2 General Orthogonal Polynomials of Two Variables

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§37.2(iv) Zeros

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2: 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

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§37.10(iii) Bernstein–Szegő Polynomials of Two Variables

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§37.10(iv) Hahn polynomials of Two Variables

… ►As an example we give the Hahn polynomials of two variables: …

3: 37.8 Jacobi Polynomials Associated with Root System $B{C}_2$

§37.8 Jacobi Polynomials Associated with Root System $B{C}_2$

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37.8.5 $$L_{\alpha ,\beta ,\gamma }{p}_{k,n}^{\alpha ,\beta ,\gamma }={\lambda }_{k,n}{p}_{k,n}^{\alpha ,\beta ,\gamma },$$

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Symbols:

$n$: nonnegative integer, $k$: nonnegative integer, $\alpha$: real parameter, $\beta$: real parameter, $\gamma$: real parameter, $p_{k,n}^{\alpha ,\beta ,\gamma }$: two-variable $p$ Jacobi polynomial associated with root system $B{C}_2$ , $L_{\alpha ,\beta ,\gamma }$: differential operator for two-variable $p$ Jacobi polynomial associated with root system $B{C}_2$ and ${\lambda }_{k,n}$: eignvalue for two-variable $p$ Jacobi polynomial associated with root system $B{C}_2$

Proved:

Koornwinder (1974b, (4.4))(proved)

Permalink:

http://dlmf.nist.gov/37.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §37.8, §37.8 and Ch.37

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37.8.11 $$P_{k,n}^{\alpha ,\beta ,\gamma }(x+y,xy)=p_{k,n}^{\alpha ,\beta ,\gamma }(x,y),$$

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Symbols:

$n$: nonnegative integer, $k$: nonnegative integer, $x$: real variable, $y$: real variable, $\alpha$: real parameter, $P_{k,n}^{\alpha ,\beta ,\gamma }$: two-variable $P$ Jacobi polynomial associated with root system $B{C}_2$ , $\beta$: real parameter, $\gamma$: real parameter and $p_{k,n}^{\alpha ,\beta ,\gamma }$: two-variable $p$ Jacobi polynomial associated with root system $B{C}_2$

Referenced by:

§37.8

Permalink:

http://dlmf.nist.gov/37.8.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §37.8, §37.8 and Ch.37

… ►More generally, the definition of the symmetric OPs $p_{k,n}^{\alpha ,\beta ,\gamma }(x,y)$ can be extended to symmetric OPs $p_{k,n}(x,y)$ for weight function $W(x,y)=w(x)w(y){(x-y)}^{2\gamma +1}$ () for any weight function $w$ on $ℝ$. Moreover, the corresponding OPs $P_{k,n}(u,v)$ as in (37.8.11) satisfy for $\gamma =\pm \frac{1}{2}$ the property that ${\{P_{k,n}\}}_{k=0}^n$ has $\frac{1}{2}(n+1)(n+2)$ real common zeros; see Schmid and Xu (1994). …

4: 37.6 Plane with Weight Function ${\mathrm{e}}^{-x^2-y^2}$

§37.6 Plane with Weight Function ${\mathrm{e}}^{-x^2-y^2}$

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37.6.1 $$⟨f,g⟩=\frac{1}{\pi }{\int }_{{ℝ}^2}f(x,y)g(x,y){\mathrm{e}}^{-x^2-y^2}dxdy,$$

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Defines:

$⟨f,g⟩$: inner product over two-dimensional Gaussian weight (locally), $f$: two-variable function on ${ℝ}^2$ (locally) and $g$: two-variable function on ${ℝ}^2$ (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $ℝ$: real line, $x$: real variable, ${ℝ}^2$: real plane and $y$: real variable

Referenced by:

§37.6

Permalink:

http://dlmf.nist.gov/37.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §37.6 and Ch.37

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37.6.3

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Defines:

$S_{m,n}(z,\overline{z})$: two-variable complex circular Hermite polynomial

Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\overline{z}$: complex conjugate, $!$: factorial (as in $n!$), $n$: nonnegative integer, $z$: complex number and $m$: nonnegative integer

Sources:

Itô (1952, §12); Intissar and Intissar (2006, Proposition 2.1(i)($\beta$))

Referenced by:

§37.12(iv), §37.6(ii), §37.6(iii)

Permalink:

http://dlmf.nist.gov/37.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §37.6(i), §37.6 and Ch.37

… ►The definition of $S_{m,n}(z,\overline{z})$ can be extended to $S_{m,n}(z_1,z_2)$, where $z_1$ and $z_2$ are two independent complex variables. … ►

37.6.15 $$\underset{\alpha \to \mathrm{\infty }}{lim}{\alpha }^{\frac{1}{2}(m+n)}{R}_{m,n}^{\alpha }({\alpha }^{-\frac{1}{2}}z,{\alpha }^{-\frac{1}{2}}\overline{z})=S_{m,n}(z,\overline{z}),$$

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Symbols:

$S_{m,n}(z,\overline{z})$: two-variable complex circular Hermite polynomial, $R_{m,n}^{\alpha }(z,\overline{z})$: two-variable complex disk polynomial, $\overline{z}$: complex conjugate, $n$: nonnegative integer, $z$: complex number and $m$: nonnegative integer

Proof sketch:

Use (18.7.22).

Referenced by:

(37.6.14)

Permalink:

http://dlmf.nist.gov/37.6.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §37.6(iv), §37.6 and Ch.37

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5: 37.7 Parabolic Biangular Region with Weight Function ${\left(1-x\right)}^{\alpha }{\left(x-y^2\right)}^{\beta }$

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

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37.7.2 $$W_{\alpha ,\beta }(x,y)={(1-x)}^{\alpha }{(x-y^2)}^{\beta },$$ $\alpha ,\beta >-1$.

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Defines:

$W_{\alpha ,\beta }$: two-variable weight function on parabolic biangular region $𝔸$ (locally)

Symbols:

$x$: real variable, $y$: real variable and $𝔸$: parabolic biangular region

Permalink:

http://dlmf.nist.gov/37.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §37.7 and Ch.37

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37.7.12 $$\left(L+n(n+\alpha +\beta +\frac{3}{2})\right)P_{2k,n+k}^{\alpha ,\beta }(x,y)=0,$$

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Symbols:

$n$: nonnegative integer, $x$: real variable, $y$: real variable and $P_{k,n}^{\alpha ,\beta }$: two-variable Jacobi polynomial on the parabolic biangular region $𝔸$

Permalink:

http://dlmf.nist.gov/37.7.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §37.7(iii), §37.7 and Ch.37

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37.7.16 $$R_{k,n}^{\alpha ,\beta }(x,y)=P_{k,n}^{\beta ,\alpha }(1-x+y^2,y)=P_{n-k}^{(\beta ,\alpha +k+\frac{1}{2})}\left(1-2x+2y^2\right){(1-x+y^2)}^{\frac{1}{2}k}{P}_k^{(\alpha ,\alpha )}\left(\frac{y}{\sqrt{1-x+y^2}}\right).$$

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Defines:

$R_{k,n}^{\alpha ,\beta }$: second two-variable Jacobi polynomial on the parabolic biangular region $𝔸$ (locally)

Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $n$: nonnegative integer, $k$: nonnegative integer, $x$: real variable, $y$: real variable, $𝔸$: parabolic biangular region and $P_{k,n}^{\alpha ,\beta }$: two-variable Jacobi polynomial on the parabolic biangular region $𝔸$

Referenced by:

(37.7.17), (37.7.21), (37.7.25), §37.7(iv), §37.7(v), §37.7(vi)

Permalink:

http://dlmf.nist.gov/37.7.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §37.7(iv), §37.7 and Ch.37

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37.7.19 $$y{R}_{k,n}^{\alpha ,\frac{1}{2},\gamma }(1-x,y^2)=\frac{{(-1)}^n{\left(1+k\right)}_{k+1}}{{\left(\alpha +1+k\right)}_{k+1}}{R}_{2k+1,n+k+1}^{\alpha ,\gamma }(x,y).$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer, $k$: nonnegative integer, $x$: real variable, $y$: real variable and $R_{k,n}^{\alpha ,\beta }$: second two-variable Jacobi polynomial on the parabolic biangular region $𝔸$

Permalink:

http://dlmf.nist.gov/37.7.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §37.7(iv), §37.7 and Ch.37

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6: 37.4 Disk with Weight Function ${\left(1-x^2-y^2\right)}^{\alpha }$

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

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37.4.2 $$W_{\alpha }(x,y)={(1-x^2-y^2)}^{\alpha },$$ $\alpha >-1$,

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Defines:

$W_{\alpha }$: two-variable weight function on $𝔻$ (locally)

Symbols:

$x$: real variable, $y$: real variable, $𝔻$: unit disk and $\alpha$: real parameter

Referenced by:

§37.4(iv), §37.6(iv)

Permalink:

http://dlmf.nist.gov/37.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §37.4(i), §37.4 and Ch.37

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37.4.3 $${⟨f,g⟩}_{\alpha }=\frac{\alpha +1}{\pi }{\int }_{𝔻}f(x,y)g(x,y)W_{\alpha }(x,y)dxdy,$$ $\alpha >-1$,

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Defines:

${⟨f,g⟩}_{\alpha }$: inner product on disk (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $x$: real variable, $y$: real variable, $𝔻$: unit disk, $W_{\alpha }$: two-variable weight function on $𝔻$ , $\alpha$: real parameter, $f$: two-variable function on $𝔻$ and $g$: two-variable function on $𝔻$

Referenced by:

§37.4(i), §37.4(iii)

Permalink:

http://dlmf.nist.gov/37.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §37.4(i), §37.4 and Ch.37

… ► … ►For both real and complex disk polynomials there is the Fourier transform pair …

7: 37.5 Quarter Plane with Weight Function $x^{\alpha }{y}^{\beta }{\mathrm{e}}^{-x-y}$

§37.5 Quarter Plane with Weight Function $x^{\alpha }{y}^{\beta }{\mathrm{e}}^{-x-y}$

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37.5.2 $$W_{\alpha ,\beta }(x,y)=x^{\alpha }{y}^{\beta }{\mathrm{e}}^{-x-y},$$ $\alpha ,\beta >-1$.

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Defines:

$W$: two-variable weight function on ${ℝ}_{+}^2$ (locally)

Symbols:

$\mathrm{e}$: base of natural logarithm, $ℝ$: real line, $x$: real variable, $y$: real variable, $\alpha$: real parameter, $\beta$: real parameter and ${ℝ}_{+}^2$: quarter plane

Referenced by:

§37.5(iii)

Permalink:

http://dlmf.nist.gov/37.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §37.5 and Ch.37

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37.5.3 $${⟨f,g⟩}_{\alpha ,\beta }=\frac{1}{\mathrm{\Gamma }\left(\alpha +1\right)\mathrm{\Gamma }\left(\beta +1\right)}{\int }_{{ℝ}_{+}^2}f(x,y)g(x,y)W_{\alpha ,\beta }(x,y)dxdy,$$ $\alpha ,\beta >-1$,

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Defines:

${⟨f,g⟩}_{\alpha ,\beta }$: inner product on quarter plane (locally), $f$: two-variable function on ${ℝ}_{+}^2$ (locally) and $g$: two-variable function on ${ℝ}_{+}^2$ (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, $ℝ$: real line, $x$: real variable, $y$: real variable, $\alpha$: real parameter, $\beta$: real parameter, ${ℝ}_{+}^2$: quarter plane and $W$: two-variable weight function on ${ℝ}_{+}^2$

Referenced by:

§37.5

Permalink:

http://dlmf.nist.gov/37.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §37.5 and Ch.37

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37.5.11 $$\underset{\gamma \to \mathrm{\infty }}{lim}{\gamma }^k{R}_{k,n}^{\alpha ,\beta ,\gamma }({\gamma }^{-1}x,{\gamma }^{-1}y)={(-1)}^k{Q}_{k,n}^{(\alpha ,\beta )}(x,y),$$

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Symbols:

: two-variable $R$-Jacobi polynomial on the triangle, $n$: nonnegative integer, $k$: nonnegative integer, $x$: real variable, $y$: real variable, $\alpha$: real parameter, $\gamma$: real parameter, $\beta$: real parameter and $Q_{k,n}^{(\alpha ,\beta )}$: two-variable Laguerre–Jacobi polynomials on ${ℝ}_{+}^2$

Permalink:

http://dlmf.nist.gov/37.5.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §37.5(iii), §37.5 and Ch.37

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8: 37.9 Jacobi Polynomials Associated with Root System $A_2$

§37.9 Jacobi Polynomials Associated with Root System $A_2$

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37.9.1 $$\omega (x,y)=-{\left(x^2+y^2+9\right)}^2+8\left(x^3-3x{y}^2\right)+108=-{\left(z\overline{z}+9\right)}^2+4\left(z^3+{\overline{z}}^3\right)+108,$$

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Defines:

$\omega$: two-variable weight function associated with Root System $A_2$ (locally)

Symbols:

$\overline{z}$: complex conjugate, $z$: complex number, $x$: real variable and $y$: real variable

Source:

Koornwinder (1974c, (3.16))

Permalink:

http://dlmf.nist.gov/37.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §37.9 and Ch.37

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37.9.3

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Defines:

$P_{m,n}^{\alpha }$: two-variable Jacobi polynomial associated with root system $A_2$ (locally)

Symbols:

$\overline{z}$: complex conjugate, $n$: nonnegative integer, $z$: complex number, $m$: nonnegative integer and $\alpha$: real parameter

Permalink:

http://dlmf.nist.gov/37.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §37.9, §37.9 and Ch.37

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37.9.4 $${\int }_{\mathrm{\Omega }}{P}_{m,n}^{\alpha }(x+\mathrm{i}y,x-\mathrm{i}y){(x-\mathrm{i}y)}^k{(x+\mathrm{i}y)}^l\omega {(x,y)}^{\alpha }dxdy=0,$$ $k,l\in {\mathbb{N}}_0$, .

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Symbols:

$dx$: differential of $x$ , $\in$: element of, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathbb{N}$: set of all positive integers, $n$: nonnegative integer, $k$: nonnegative integer, $l$: nonnegative integer, $m$: nonnegative integer, ${\mathbb{N}}_0$: set of all nonnegative integers, $x$: real variable, $y$: real variable, $\omega$: two-variable weight function associated with Root System $A_2$ , $\mathrm{\Omega }$: region bounded by deltoid associated with Root System $A_2$ , $\alpha$: real parameter and $P_{m,n}^{\alpha }$: two-variable Jacobi polynomial associated with root system $A_2$

Permalink:

http://dlmf.nist.gov/37.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §37.9, §37.9 and Ch.37

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37.9.5 $${\int }_{\mathrm{\Omega }}{P}_{m,n}^{\alpha }(x+\mathrm{i}y,x-\mathrm{i}y)\overline{P_{k,l}^{\alpha }(x+\mathrm{i}y,x-\mathrm{i}y)}\omega {(x,y)}^{\alpha }dxdy=0,$$ $\alpha >-\frac{5}{6}$, $(m,n)\ne (k,l)$.

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Symbols:

$\overline{z}$: complex conjugate, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $(a,b)$: open interval, $n$: nonnegative integer, $k$: nonnegative integer, $l$: nonnegative integer, $m$: nonnegative integer, $x$: real variable, $y$: real variable, $\omega$: two-variable weight function associated with Root System $A_2$ , $\mathrm{\Omega }$: region bounded by deltoid associated with Root System $A_2$ , $\alpha$: real parameter and $P_{m,n}^{\alpha }$: two-variable Jacobi polynomial associated with root system $A_2$

Proved:

Koornwinder (1974c, Corollary 6.4)(proved)

Referenced by:

§37.9

Permalink:

http://dlmf.nist.gov/37.9.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §37.9, §37.9 and Ch.37

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9: 37.20 Mathematical Applications

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Numerical Integration and Interpolation

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10: 37.1 Notation

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$x$, $y$	real variables.
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$P,Q$	polynomials of two variables.

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