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1: 37.2 General Orthogonal Polynomials of Two Variables

§37.2 General Orthogonal Polynomials of Two Variables

… ► ${\mathrm{\Pi }}_n$ is the space of polynomials of degree $\le n$. …The space ${𝒱}_n$ of orthogonal polynomials of degree $n$ consists of all $P\in {\mathrm{\Pi }}_n$ such that ${⟨P,Q⟩}_W=0$ for all $Q\in {\mathrm{\Pi }}_{n-1}$ ($n>0$, otherwise ${𝒱}_0={\mathrm{\Pi }}_0$). … ►

§37.2(i) Bases of ${𝒱}_n$

… ► …

2: 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 ${\left(1-x\right)}^{\alpha }{\left(x-y^2\right)}^{\beta }$

… ►

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

ⓘ

Symbols:

$n$: nonnegative integer, $x$: real variable, $y$: real variable, $\alpha$: real parameter, $\beta$: real parameter, $P_{k,n}^{\alpha ,\beta }$: two-variable Jacobi polynomial on the parabolic biangular region $𝔸$ and $L$: second order partial differential operator in two variables

Permalink:

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

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

… ►Thus ${\{P_{2k,n+k}^{\alpha ,\beta }\}}_{k=0}^n$ is an orthogonal basis of the eigenspace, denoted by ${𝒲}_n^{\alpha ,\beta }$, of $L$ for eigenvalue $-n(n+\alpha +\beta +\frac{3}{2})$, and ${\{P_{2k+1,n+k+1}^{\alpha ,\beta }\}}_{k=0}^n$ is an orthogonal basis of the eigenspace ${𝒲}_{n+\frac{1}{2}}^{\alpha ,\beta }$ for eigenvalue $-(n+\frac{1}{2})(n+\alpha +\beta +2)$. ►Within ${𝒲}_n^{\alpha ,\beta }$ and ${𝒲}_{n+\frac{1}{2}}^{\alpha ,\beta }$ the above special bases can be obtained as joint eigenfunctions of the PDO in (37.7.14) and of another PDO commuting with the first one. … ►In fact, ${\{R_{2k,n+k}^{\alpha ,\beta }\}}_{k=0}^n$ is another orthogonal basis of ${𝒲}_n^{\alpha ,\beta }$ and ${\{R_{2k+1,n+k+1}^{\alpha ,\beta }\}}_{k=0}^n$ is another orthogonal basis of ${𝒲}_{n+\frac{1}{2}}^{\alpha ,\beta }$. …

3: 37.6 Plane with Weight Function ${\mathrm{e}}^{-x^2-y^2}$

… ►The OPs of degree $n$ with respect to the inner product (37.6.1) form the space ${𝒱}_n$. The spaces ${𝒱}_n$ are eigenspaces of a second order partial differential operator, see (37.6.12). … ►There is an obvious orthogonal basis of ${𝒱}_n$ consisting of products of Hermite polynomials: … ►The spaces ${𝒱}_n$ are eigenspaces of a second order partial differential operator: … ►Special bases of ${𝒱}_n$ can be obtained as joint eigenfunctions of the PDO in (37.6.12) and of another PDO commuting with the first one. …

4: 37.10 Other Orthogonal Polynomials of Two Variables

§37.10 Other Orthogonal Polynomials of Two Variables

… ►

§37.10(ii) Orthogonal Polynomials on an Annulus

… ►

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

… ►

§37.10(iv) Hahn polynomials of Two Variables

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

5: 37.8 Jacobi Polynomials Associated with Root System $B{C}_2$

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

… ►

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

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See also:

Annotations for §37.8, §37.8 and Ch.37

… ►

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

ⓘ

Symbols:

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

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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). …

6: 37.5 Quarter Plane with Weight Function $x^{\alpha }{y}^{\beta }{\mathrm{e}}^{-x-y}$

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

… ►

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:

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See also:

Annotations for §37.5 and Ch.37

… ►

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 integrable function on ${ℝ}_{+}^2$ (locally) and $g$: two-variable integrable 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

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Annotations for §37.5 and Ch.37

… ►The spaces ${𝒱}_n^{\alpha ,\beta }$ are eigenspaces of a second order partial differential operator, see (37.5.8). … ►

37.5.8 $$[x{D}_{xx}+y{D}_{yy}+(1+\alpha -x)D_x+(1+\beta -y)D_y]u(x,y)=-nu(x,y),$$ $u\in {𝒱}_n^{\alpha ,\beta }$.

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Defines:

$u$: OP in two variables on the quarter plane (locally)

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\in$: element of, ${𝒱}_n^{\alpha ,\beta }$: vector space of two-variable orthogonal polynomials on the quarter plane, $n$: nonnegative integer, $x$: real variable, $y$: real variable, $\alpha$: real parameter and $\beta$: real parameter

Referenced by:

item 3, §37.5(ii), §37.5

Permalink:

http://dlmf.nist.gov/37.5.E8

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

…

7: 37.4 Disk with Weight Function ${\left(1-x^2-y^2\right)}^{\alpha }$

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

… ►

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

ⓘ

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

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

… ►

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

ⓘ

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 integrable function on $𝔻$ and $g$: two-variable integrable function on $𝔻$

Referenced by:

§37.4(i), §37.4(iii)

Permalink:

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

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

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

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

§37.9 Jacobi Polynomials Associated with Root System $A_2$

… ►

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, $y$: real variable and $A_2$: root system

Source:

Koornwinder (1974c, (3.16))

Permalink:

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

Encodings:

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See also:

Annotations for §37.9 and Ch.37

… ►

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, $A_2$: root system and $\alpha$: real parameter

Permalink:

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

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Annotations for §37.9, §37.9 and Ch.37

►

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

ⓘ

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:

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See also:

Annotations for §37.9, §37.9 and Ch.37

… ►

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

ⓘ

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:

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Annotations for §37.9, §37.9 and Ch.37

…

9: 37.20 Mathematical Applications

… ►

Numerical Integration and Interpolation

…

10: 37.3 Triangular Region with Weight Function

… ►

§37.3(i) Orthogonal Decomposition

… ►

37.3.1

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Defines:

: weight function in two-variables on $\mathrm{△}$ (locally)

Symbols:

$\mathrm{△}$: triangular region, $x$: real variable, $y$: real variable, : real parameter ($>-1$), : real parameter ($>-1$) and : real parameter ($>-1$)

Referenced by:

§37.5(iii)

Permalink:

http://dlmf.nist.gov/37.3.E1

Encodings:

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See also:

Annotations for §37.3(i), §37.3 and Ch.37

… ►

37.3.2 ,

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Defines:

: inner product on triangular region (locally), : real parameter ($>-1$) (locally), : real parameter ($>-1$) (locally) and : real parameter ($>-1$) (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{△}$: triangular region, $x$: real variable, $y$: real variable, : weight function in two-variables on $\mathrm{△}$ , $f$: two-variable integrable function on $\mathrm{△}$ and $g$: two-variable integrable function on $\mathrm{△}$

Referenced by:

(37.3.18), (37.3.19), (37.3.20), §37.3(i)

Permalink:

http://dlmf.nist.gov/37.3.E2

Encodings:

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See also:

Annotations for §37.3(i), §37.3 and Ch.37

… ► … ►

37.3.15 ,

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\in$: element of, : vector space of two-variable orthogonal polynomials on the triangular region, $n$: nonnegative integer, $z$: complex number, $x$: real variable, $y$: real variable, : weight function in two-variables on $\mathrm{△}$ , $u$: OP in two variables on the triangular region, : real parameter ($>-1$), : real parameter ($>-1$) and : real parameter ($>-1$)

Proof sketch:

By direct computation the left-hand side of the present equality equals the left-hand side of (37.3.14).

Permalink:

http://dlmf.nist.gov/37.3.E15

Encodings:

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See also:

Annotations for §37.3(iv), §37.3 and Ch.37

…