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1: 37.1 Notation

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$d$	positive integer, usually $\ge 2$.
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${ℝ}^d$	$d$-dimensional Euclidean space.
$𝐱,𝐲$	$(x_1,\mathrm{\dots },x_d),(y_1,\mathrm{\dots },y_d)\in {ℝ}^d$.
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2: 37.17 Hermite Polynomials on ${ℝ}^d$

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37.17.1 $$⟨f,g⟩={\pi }^{-\frac{1}{2}d}{\int }_{{ℝ}^d}f(𝐱)g(𝐱){\mathrm{e}}^{-{\Vert 𝐱\Vert }^2}d𝐱,$$

ⓘ

Defines:

$⟨f,g⟩$: inner product for $d$-variable Hermite polynomials (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, $\Vert 𝐯\Vert$: vector norm ($l$²), $d$: positive integer, usually $\ge 2$ , ${ℝ}^d$: $d$-dimensional Euclidean space and $𝐱$: vector in $d$-dimensional Euclidean space

Referenced by:

§37.17

Permalink:

http://dlmf.nist.gov/37.17.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §37.17, Part 37.III and Ch.37

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37.17.2 $$H_{𝝂}(𝐱)=H_{{\nu }_1}\left(x_1\right)\mathrm{\dots }{H}_{{\nu }_d}\left(x_d\right),$$ $|𝝂|=n$,

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space and $x$: real variable

Referenced by:

§37.17(v)

Permalink:

http://dlmf.nist.gov/37.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §37.17(i), §37.17, Part 37.III and Ch.37

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37.17.5 $$\sum _{|𝝂|=n}\frac{H_{𝝂}(𝐱)}{𝝂!}{𝐲}^{𝝂}=\frac{1}{n!}{H}_n\left(⟨𝐱,𝐲⟩\right),$$ $\Vert 𝐲\Vert =1$,

ⓘ

Symbols:

$H_n\left(x\right)$: Hermite polynomial, $!$: factorial (as in $n!$), $⟨𝐮,𝐯⟩$: inner product over vectors, $\Vert 𝐯\Vert$: vector norm ($l$²), $n$: nonnegative integer, $𝐱$: vector in $d$-dimensional Euclidean space and $𝐲$: vector in $d$-dimensional Euclidean space

Proof sketch:

Take $\Vert 𝐲\Vert =1$ in (37.17.4).

Permalink:

http://dlmf.nist.gov/37.17.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §37.17(i), §37.17(i), §37.17, Part 37.III and Ch.37

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37.17.6 $$\sum _{𝝂\in {\mathbb{N}}_0^d}\frac{H_{𝝂}(𝐱)}{𝝂!}{𝐲}^{𝝂}=\exp \left(2⟨𝐱,𝐲⟩-{\Vert 𝐲\Vert }^2\right),$$ $𝐲\in {ℝ}^d$.

ⓘ

Symbols:

$\in$: element of, $\exp z$: exponential function, $!$: factorial (as in $n!$), $⟨𝐮,𝐯⟩$: inner product over vectors, $\mathbb{N}$: set of all positive integers, $ℝ$: real line, $\Vert 𝐯\Vert$: vector norm ($l$²), $d$: positive integer, usually $\ge 2$ , ${ℝ}^d$: $d$-dimensional Euclidean space, $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space and ${\mathbb{N}}_0$: set of all nonnegative integers

Proof sketch:

This follows from (18.12.15).

Permalink:

http://dlmf.nist.gov/37.17.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §37.17(i), §37.17(i), §37.17, Part 37.III and Ch.37

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37.17.15 $$P_{𝝂}(𝐱)=H_{𝝂}\left(𝐁𝐱\right),$$ $|𝝂|=n$.

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Symbols:

$n$: nonnegative integer, $P$: polynomial of two variables and $𝐱$: vector in $d$-dimensional Euclidean space

Permalink:

http://dlmf.nist.gov/37.17.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §37.17(vi), §37.17, Part 37.III and Ch.37

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3: 37.13 General Orthogonal Polynomials of $d$ Variables

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37.13.1 $${⟨f,g⟩}_W={\int }_{\mathrm{\Omega }}f(𝐱)g(𝐱)W(𝐱)d𝐱$$

ⓘ

Defines:

${⟨f,g⟩}_W$: inner product on $d$-dimensional Euclidean space (locally)

Symbols:

$dx$: differential of $x$ , $\int$: integral, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space and $W$: weight function in $d$-variables

Referenced by:

§37.13

Permalink:

http://dlmf.nist.gov/37.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §37.13, Part 37.III and Ch.37

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37.13.4 $${\int }_{\mathrm{\Omega }}{𝐑}_n(𝐱,𝐲)Q(𝐲)W(𝐲)d𝐲=Q(𝐱),$$ $Q\in {𝒱}_n^d$, $𝐱\in {ℝ}^d$.

ⓘ

Symbols:

$dx$: differential of $x$ , $\in$: element of, $\int$: integral, $ℝ$: real line, $n$: nonnegative integer, $Q$: polynomial of two variables, $d$: positive integer, usually $\ge 2$ , ${ℝ}^d$: $d$-dimensional Euclidean space, $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space, $W$: weight function in $d$-variables, ${𝒱}_n^d$: vector space of $d$-variable polynomials of degree $n$ and ${𝐑}_n$: reproducing kernel

Permalink:

http://dlmf.nist.gov/37.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §37.13, §37.13, Part 37.III and Ch.37

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37.13.5 $${𝐑}_n(𝐱,𝐲)=\sum _{|𝝂|=n}\frac{P_{𝝂}(𝐱)P_{𝝂}(𝐲)}{{⟨P_{𝝂},P_{𝝂}⟩}_W},$$ $𝐱,𝐲\in {ℝ}^d$.

ⓘ

Symbols:

$\in$: element of, $ℝ$: real line, $n$: nonnegative integer, $P$: polynomial of two variables, ${ℝ}^d$: $d$-dimensional Euclidean space, $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space, $W$: weight function in $d$-variables, ${⟨f,g⟩}_W$: inner product on $d$-dimensional Euclidean space and ${𝐑}_n$: reproducing kernel

Referenced by:

§37.14(v), §37.15(vi)

Permalink:

http://dlmf.nist.gov/37.13.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §37.13, §37.13, Part 37.III and Ch.37

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37.13.6 $${𝐏}_z(𝐱,𝐲)=\sum _{n=0}^{\mathrm{\infty }}{𝐑}_n(𝐱,𝐲)z^n,$$ ,

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Symbols:

$(a,b)$: open interval, $n$: nonnegative integer, $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space, $z$: complex number and ${𝐑}_n$: reproducing kernel

Referenced by:

§37.13, §37.16(ii), §37.17(iv)

Permalink:

http://dlmf.nist.gov/37.13.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §37.13, §37.13, Part 37.III and Ch.37

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37.13.13 $$w(𝐱)=x_1^{{\alpha }_1}\mathrm{\cdots }{x}_c^{{\alpha }_c}{\mathrm{e}}^{-(x_1+\mathrm{\cdots }+x_c)}{\mathrm{e}}^{-(x_{c+1}^2+\mathrm{\cdots }+x_d^2)},$$ $𝐱\in {ℝ}_{+}^c\times {ℝ}^{d-c}$.

ⓘ

Symbols:

$\in$: element of, $\mathrm{e}$: base of natural logarithm, $ℝ$: real line, $d$: positive integer, usually $\ge 2$ , ${ℝ}_{+}$: positive real line, ${ℝ}^d$: $d$-dimensional Euclidean space, $𝐱$: vector in $d$-dimensional Euclidean space and $x$: real variable

Permalink:

http://dlmf.nist.gov/37.13.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §37.13(ii), §37.13, Part 37.III and Ch.37

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4: 37.15 Orthogonal Polynomials on the Ball

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37.15.1

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Defines:

${𝔹}^d$: unit ball (locally)

Symbols:

$\in$: element of, $ℝ$: real line, $\Vert 𝐯\Vert$: vector norm ($l$²), ${ℝ}^d$: $d$-dimensional Euclidean space and $𝐱$: vector in $d$-dimensional Euclidean space

Permalink:

http://dlmf.nist.gov/37.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §37.15(i), §37.15, Part 37.III and Ch.37

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37.15.2 $$W_{\alpha }(𝐱)={\left(1-{\Vert 𝐱\Vert }^2\right)}^{\alpha },$$ $\alpha >-1$,

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Defines:

$W$: weight function in $d$-variables (locally)

Symbols:

$\Vert 𝐯\Vert$: vector norm ($l$²), $d$: positive integer, usually $\ge 2$ and $𝐱$: vector in $d$-dimensional Euclidean space

Referenced by:

§37.15(iii), §37.18(ii), §37.18(iii)

Permalink:

http://dlmf.nist.gov/37.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §37.15(i), §37.15, Part 37.III and Ch.37

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37.15.13 $${\left(1-2⟨𝐱,𝐲⟩+{\Vert 𝐲\Vert }^2\right)}^{-\alpha -\frac{1}{2}d}=\sum _{𝝂\in {\mathbb{N}}_0^d}\frac{2^{|𝝂|}{\left(\alpha +\frac{1}{2}d\right)}_{|𝝂|}}{𝝂!}{𝐲}^{𝝂}{V}_{𝝂}^{(\alpha +\frac{1}{2})}(𝐱),$$ $𝐲\in {ℝ}^d$, .

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\in$: element of, $!$: factorial (as in $n!$), $⟨𝐮,𝐯⟩$: inner product over vectors, $\mathbb{N}$: set of all positive integers, $ℝ$: real line, $\Vert 𝐯\Vert$: vector norm ($l$²), $d$: positive integer, usually $\ge 2$ , ${ℝ}^d$: $d$-dimensional Euclidean space, $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space and ${\mathbb{N}}_0$: set of all nonnegative integers

Proved:

Dunkl and Xu (2014, Proposition 5.2.3); There the polynomial $V_{𝝂}^{(\alpha +\frac{1}{2})}$ is denoted by $V_{\alpha }$.(proved)

Referenced by:

(37.15.14)

Permalink:

http://dlmf.nist.gov/37.15.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §37.15(iv), §37.15(iv), §37.15, Part 37.III and Ch.37

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37.15.22 $${𝐱}^{\mathit{ϵ}}{P}_{𝝂}^{-\frac{\mathrm{𝟏}}{\mathrm{𝟐}}+\mathit{ϵ},\alpha }({𝐱}^2)=\mathrm{const}.C_{2𝝂+\mathit{ϵ}}^{(\alpha +\frac{1}{2})}(𝐱),$$ $\mathit{ϵ}\in {\{0,1\}}^d$.

ⓘ

Symbols:

$\in$: element of, $P$: polynomial of two variables, $d$: positive integer, usually $\ge 2$ and $𝐱$: vector in $d$-dimensional Euclidean space

Permalink:

http://dlmf.nist.gov/37.15.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §37.15(vii), §37.15, Part 37.III and Ch.37

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37.15.23 $${𝐱}^{\mathit{ϵ}}{V}_{𝝂}^{-\frac{\mathrm{𝟏}}{\mathrm{𝟐}}+\mathit{ϵ},\alpha }({𝐱}^2)=\mathrm{const}.V_{2𝝂+\mathit{ϵ}}^{(\alpha +\frac{1}{2})}(𝐱),$$ $\mathit{ϵ}\in {\{0,1\}}^d$,

ⓘ

Symbols:

$\in$: element of, $d$: positive integer, usually $\ge 2$ and $𝐱$: vector in $d$-dimensional Euclidean space

Permalink:

http://dlmf.nist.gov/37.15.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §37.15(vii), §37.15, Part 37.III and Ch.37

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5: 37.16 Orthogonal Polynomials on the Hyperoctant

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37.16.1 $${ℝ}_{+}^d=\{𝐱\in {ℝ}^d\mid x_1,\mathrm{\dots },x_d>0\}$$

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Defines:

${ℝ}_{+}^d$: hyperoctant (locally)

Symbols:

$\in$: element of, $ℝ$: real line, $d$: positive integer, usually $\ge 2$ , ${ℝ}^d$: $d$-dimensional Euclidean space, $𝐱$: vector in $d$-dimensional Euclidean space and $x$: real variable

Permalink:

http://dlmf.nist.gov/37.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §37.16, Part 37.III and Ch.37

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37.16.2 $$W_{𝜶}(𝐱)={𝐱}^{𝜶}{\mathrm{e}}^{-|𝐱|},$$ ${\alpha }_1,\mathrm{\dots },{\alpha }_d>-1$.

ⓘ

Defines:

$W$: weight function in $d$-variables (locally)

Symbols:

$\mathrm{e}$: base of natural logarithm, $d$: positive integer, usually $\ge 2$ and $𝐱$: vector in $d$-dimensional Euclidean space

Permalink:

http://dlmf.nist.gov/37.16.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §37.16, Part 37.III and Ch.37

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37.16.3 $${⟨f,g⟩}_{𝜶}=\frac{1}{{\prod }_{\mathrm{\ell }=1}^d\mathrm{\Gamma }\left({\alpha }_{\mathrm{\ell }}+1\right)}{\int }_{{ℝ}_{+}^d}f(𝐱)g(𝐱)W_{𝜶}(𝐱)d𝐱,$$ ${\alpha }_1,\mathrm{\dots },{\alpha }_d>-1$,

ⓘ

Defines:

${⟨f,g⟩}_{𝜶}$: inner product on hyperoctant (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, $ℝ$: real line, $d$: positive integer, usually $\ge 2$ , $\mathrm{\ell }$: positive integer, $𝐱$: vector in $d$-dimensional Euclidean space, ${ℝ}_{+}^d$: hyperoctant and $W$: weight function in $d$-variables

Referenced by:

§37.16

Permalink:

http://dlmf.nist.gov/37.16.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §37.16, Part 37.III and Ch.37

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37.16.7 $${𝐏}_z^{\alpha }(𝐱,𝐲)=\sum _{n=0}^{\mathrm{\infty }}{𝐑}_n^{𝜶}(𝐱,𝐲)z^n={(1-z)}^{-1}\exp \left(\frac{z(|𝐱|+|𝐲|)}{z-1}\right)\prod _{\mathrm{\ell }=1}^d\frac{\mathrm{\Gamma }\left({\alpha }_{\mathrm{\ell }}+1\right)}{{\left(x_{\mathrm{\ell }}{y}_{\mathrm{\ell }}z\right)}^{\frac{1}{2}{\alpha }_{\mathrm{\ell }}}}{I}_{{\alpha }_{\mathrm{\ell }}}\left(\frac{2\sqrt{x_{\mathrm{\ell }}{y}_{\mathrm{\ell }}z}}{1-z}\right),$$ , $𝐱,𝐲\in {ℝ}_{+}^d$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\in$: element of, $\exp z$: exponential function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $(a,b)$: open interval, $ℝ$: real line, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $\mathrm{\ell }$: positive integer, $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space, $z$: complex number, ${ℝ}_{+}^d$: hyperoctant, ${𝐑}_n$: reproducing kernel, $x$: real variable and $y$: real variable

Proved:

Dunkl and Xu (2014, Proposition 8.4.8); There set $V_{\kappa }=\mathrm{id}$.(proved)

Permalink:

http://dlmf.nist.gov/37.16.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §37.16(ii), §37.16, Part 37.III and Ch.37

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37.16.8 $$\underset{\beta \to \mathrm{\infty }}{lim}{P}_{𝝂}^{𝜶,\beta }({\beta }^{-1}𝐱)={(-1)}^{|𝝂|}{L}_{{\nu }_1}^{({\alpha }_1)}\left(x_1\right)\mathrm{\dots }{L}_{{\nu }_d}^{({\alpha }_d)}\left(x_d\right),$$

ⓘ

Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $P$: polynomial of two variables, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space and $x$: real variable

Permalink:

http://dlmf.nist.gov/37.16.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §37.16(iii), §37.16, Part 37.III and Ch.37

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6: 37.18 Orthogonal Polynomials on Quadratic Domains

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37.18.1 $$W(𝐱,t)=w_1(t)w_2\left(\frac{𝐱}{\varphi (t)}\right).$$

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Defines:

$W$: weight function in $d$-variables (locally)

Symbols:

$d$: positive integer, usually $\ge 2$ and $𝐱$: vector in $d$-dimensional Euclidean space

Referenced by:

§37.18(ii), §37.18(iii)

Permalink:

http://dlmf.nist.gov/37.18.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §37.18(i), §37.18, Part 37.III and Ch.37

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37.18.2 $$⟨f,g⟩=c_W{\int }_{{𝕍}^{d+1}}f(𝐱,t)g(𝐱,t)W(𝐱,t)dxdt,$$

ⓘ

Defines:

$⟨f,g⟩$: inner product on quadratic domain (locally)

Symbols:

$dx$: differential of $x$ , $\int$: integral, $(a,b)$: open interval, $𝐱$: vector in $d$-dimensional Euclidean space, ${𝕍}^{d+1}$: quadratic domain, $W$: weight function in $d$-variables and $x$: real variable

Permalink:

http://dlmf.nist.gov/37.18.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §37.18(i), §37.18, Part 37.III and Ch.37

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37.18.3 $$Q_{m,𝐤}^n(𝐱,t)=q_{n-m}^{(m)}(t)\varphi {(t)}^m{P}_{𝐤}^m\left(\frac{𝐱}{\varphi (t)}\right),$$ $0\le m\le n,|𝐤|=m$

ⓘ

Symbols:

$m$: nonnegative integer, $n$: nonnegative integer, $P$: polynomial of two variables, $Q$: polynomial of two variables, $d$: positive integer, usually $\ge 2$ and $𝐱$: vector in $d$-dimensional Euclidean space

Source:

Olver and Xu (2020, (5.2)); Here $q_{n-m}^{(2m+d)}(t)$ should be $q_{n-m}^{(m)}(t)$.

Referenced by:

§37.18(iii)

Permalink:

http://dlmf.nist.gov/37.18.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §37.18(i), §37.18(i), §37.18, Part 37.III and Ch.37

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37.18.9 $${𝐑}_n((𝐱,t),(𝐲,s))=c_{\mu ,\gamma }{\int }_{{[-1,1]}^3}{Z}_{2n}^{(2\mu +\gamma +d)}\left(\zeta (𝐱,t,𝐲,s;𝐯)\right){(1-u^2)}^{\mu -1}{(1-v_1^2)}^{\mu +\frac{d-3}{2}}{(1-v_2^2)}^{\gamma -\frac{1}{2}}d𝐯du,$$ $d\ge 2,\mu >0,\gamma >-\frac{1}{2}$,

ⓘ

Symbols:

$[a,b]$: closed interval, $dx$: differential of $x$ , $\int$: integral, $(a,b)$: open interval, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space and ${𝐑}_n$: reproducing kernel

Proved:

Xu (2020, Theorem 4.3)(proved)

Referenced by:

§37.18(ii)

Permalink:

http://dlmf.nist.gov/37.18.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §37.18(ii), §37.18(ii), §37.18, Part 37.III and Ch.37

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37.18.10 $$\zeta (𝐱,t,𝐲,s;𝐯)=v_1\sqrt{\frac{1}{2}\left(st+⟨𝐱,𝐲⟩+x_{d+1}{y}_{d+1}\right)}+v_2\sqrt{1-t}\sqrt{1-s}$$

ⓘ

Symbols:

$⟨𝐮,𝐯⟩$: inner product over vectors, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/37.18.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §37.18(ii), §37.18(ii), §37.18, Part 37.III and Ch.37

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7: 37.14 Orthogonal Polynomials on the Simplex

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37.14.1 $${\mathrm{△}}^d=\{𝐱\in {ℝ}^d\mid x_1>0,\mathrm{\dots },x_d>0,1-|𝐱|>0\}$$

ⓘ

Defines:

${\mathrm{△}}^d$: simplex (locally)

Symbols:

$\in$: element of, $ℝ$: real line, $d$: positive integer, usually $\ge 2$ , ${ℝ}^d$: $d$-dimensional Euclidean space, $𝐱$: vector in $d$-dimensional Euclidean space and $x$: real variable

Permalink:

http://dlmf.nist.gov/37.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §37.14(i), §37.14, Part 37.III and Ch.37

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37.14.2 $𝜶=({\alpha }_1,\mathrm{\dots },{\alpha }_{d+1})$, $𝐱\in {\mathrm{△}}^d$

ⓘ

Defines:

${\alpha }_{\mathrm{\ell }}$: real parameter ($>-1$) (locally)

Symbols:

$\in$: element of, $d$: positive integer, usually $\ge 2$ , $\mathrm{\ell }$: positive integer, $𝐱$: vector in $d$-dimensional Euclidean space, ${\mathrm{△}}^d$: simplex and $x$: real variable

Permalink:

http://dlmf.nist.gov/37.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §37.14(i), §37.14, Part 37.III and Ch.37

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37.14.3 $|𝜶|={\alpha }_1+\mathrm{\cdots }+{\alpha }_{d+1}$, ${\alpha }_1,\mathrm{\dots },{\alpha }_{d+1}>-1$,

ⓘ

Defines:

${⟨f,g⟩}_{𝜶}$: inner product on $d$-dimensional simplex (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, $d$: positive integer, usually $\ge 2$ , $\mathrm{\ell }$: positive integer, $𝐱$: vector in $d$-dimensional Euclidean space, ${\mathrm{△}}^d$: simplex and ${\alpha }_{\mathrm{\ell }}$: real parameter ($>-1$)

Referenced by:

§37.14(i)

Permalink:

http://dlmf.nist.gov/37.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §37.14(i), §37.14, Part 37.III and Ch.37

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37.14.7 $$P_{𝝂}^{𝜶}(𝐱)=\prod _{j=1}^d{\left(1-(x_1+\mathrm{\cdots }+x_{j-1})\right)}^{{\nu }_j}{P}_{{\nu }_j}^{({\alpha }_{j+1}+\mathrm{\cdots }+{\alpha }_{d+1}+2({\nu }_{j+1}+\mathrm{\cdots }+{\nu }_d)+d-j,{\alpha }_j)}\left(\frac{2x_j}{1-(x_1+\mathrm{\cdots }+x_{j-1})}-1\right).$$

ⓘ

Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $j$: nonnegative integer, $P$: polynomial of two variables, $d$: positive integer, usually $\ge 2$ , $\mathrm{\ell }$: positive integer, $𝐱$: vector in $d$-dimensional Euclidean space and $x$: real variable

Source:

Dunkl and Xu (2014, Proposition 5.3.1); replace ${\alpha }_{\mathrm{\ell }}$ by ${\alpha }_{\mathrm{\ell }}+\frac{1}{2}$ ($\mathrm{\ell }=1,\mathrm{\dots },d+1$)

Referenced by:

§37.14(ii), §37.15(vii), §37.16(iii), §37.19(v)

Permalink:

http://dlmf.nist.gov/37.14.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §37.14(ii), §37.14, Part 37.III and Ch.37

… ►

37.14.13 $${𝐑}_n^{𝜶}(𝐱,𝐲)={\int }_{{[-1,1]}^{d+1}}{Z}_{2n}^{(|𝜶|+d)}\left(\xi (𝐱,𝐲,𝐭)\right)\frac{{\prod }_{\mathrm{\ell }=1}^{d+1}{(1-t_{\mathrm{\ell }}^2)}^{{\alpha }_{\mathrm{\ell }}-\frac{1}{2}}}{{\int }_{{[-1,1]}^{d+1}}{\prod }_{\mathrm{\ell }=1}^{d+1}{(1-t_{\mathrm{\ell }}^2)}^{{\alpha }_{\mathrm{\ell }}-1/2}d𝐭}d𝐭,,$$ $\xi (𝐱,𝐲,𝐭)=\sqrt{x_1{y}_1}{t}_1+\mathrm{\cdots }+\sqrt{x_{d+1}{y}_{d+1}}{t}_{d+1}$ $x_{d+1}=1-|𝐱|,y_{d+1}=1-|𝐲|,{\alpha }_{\mathrm{\ell }}>-\frac{1}{2}$.

ⓘ

Symbols:

$[a,b]$: closed interval, $dx$: differential of $x$ , $\int$: integral, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $\mathrm{\ell }$: positive integer, $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space, ${𝐑}_n$: reproducing kernel, $x$: real variable and $y$: real variable

Proved:

Dunkl and Xu (2014, Theorem 5.3.4); There replace ${\alpha }_{\mathrm{\ell }}$ by ${\alpha }_{\mathrm{\ell }}+\frac{1}{2}$ ($\mathrm{\ell }=1,\mathrm{\dots },d+1$).(proved)

Referenced by:

§37.14(v), §37.3(i)

Permalink:

http://dlmf.nist.gov/37.14.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §37.14(v), §37.14, Part 37.III and Ch.37

…

8: 37.12 Orthogonal Polynomials on Quadratic Surfaces

… ►

37.12.2 $$S_{m,\mathrm{\ell }}^n(x,t)=q_{n-m}^{(m)}(t)\varphi {(t)}^m{Y}_{\mathrm{\ell }}^m\left(\frac{𝐱}{\varphi (t)}\right),$$ $0\le m\le n,\mathrm{\hspace{0.17em}\hspace{0.17em}1}\le \mathrm{\ell }\le N_m^d$.

ⓘ

Symbols:

$(a,b)$: open interval, $m$: nonnegative integer, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $\mathrm{\ell }$: positive integer, $𝐱$: vector in $d$-dimensional Euclidean space and $x$: real variable

Proved:

Olver and Xu (2020, Theorem 4.3)(proved)

Referenced by:

§37.12(i), §37.12(i), §37.12(ii), §37.12(iii)

Permalink:

http://dlmf.nist.gov/37.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §37.12(i), §37.12(i), §37.12, Part 37.III and Ch.37

… ►

37.12.5 $${𝐑}_n((x,t),(y,s))=\sum _{m=0}^n\sum _{\mathrm{\ell }=1}^{N_m^d}\frac{S_{m,\mathrm{\ell }}^n(𝐱,t)\overline{S_{m,\mathrm{\ell }}^n(𝐲,s)}}{h_{m,n}},$$ $(𝐱,t),(𝐲,s)\in {𝕍}_0^{d+1}$.

ⓘ

Symbols:

$\overline{z}$: complex conjugate, $\in$: element of, $(a,b)$: open interval, $m$: nonnegative integer, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $\mathrm{\ell }$: positive integer, $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space, ${𝕍}_0^{d+1}$: quadratic surface, ${𝐑}_n$: reproducing kernel, $x$: real variable and $y$: real variable

Referenced by:

§37.12(i), §37.12(ii), §37.12(iii)

Permalink:

http://dlmf.nist.gov/37.12.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §37.12(i), §37.12(i), §37.12, Part 37.III and Ch.37

… ►

37.12.7 $$S_{\mathrm{\ell },m}^n(𝐱,t;\beta ,\gamma )=R_{n-m}^{(\gamma ,\beta +2m+d-1)}\left(2t-1\right)t^m{Y}_{\mathrm{\ell }}^m\left(\frac{𝐱}{t}\right).$$

ⓘ

Symbols:

$R_n^{(\alpha ,\beta )}\left(x\right)$: Standardized Jacobi polynomial, $m$: nonnegative integer, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $\mathrm{\ell }$: positive integer and $𝐱$: vector in $d$-dimensional Euclidean space

Proved:

Xu (2020, Proposition 7.1)(proved)

Referenced by:

§37.12(ii), §37.12(iii), §37.12(iv)

Permalink:

http://dlmf.nist.gov/37.12.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §37.12(ii), §37.12, Part 37.III and Ch.37

… ►

37.12.9 $${𝐑}_n(w_{-1,\gamma };(𝐱,t),(𝐲,s))=\frac{{\int }_{{[-1,1]}^2}{Z}_{2n}^{(\gamma +d)}\left(\zeta (𝐱,t,𝐲,s;v)\right){(1-v_1^2)}^{\frac{d-4}{2}}{(1-v_2^2)}^{\gamma -\frac{1}{2}}{dv}_1{dv}_2}{{\int }_{{[-1,1]}^2}{(1-v_1^2)}^{\frac{d-4}{2}}{(1-v_2^2)}^{\gamma -\frac{1}{2}}{dv}_1{dv}_2},$$ $d>2,\gamma >-\frac{1}{2}$,

ⓘ

Symbols:

$[a,b]$: closed interval, $dx$: differential of $x$ , $\int$: integral, $(a,b)$: open interval, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space and ${𝐑}_n$: reproducing kernel

Proved:

Xu (2020, Theorem 8.2)(proved)

Referenced by:

§37.12(ii)

Permalink:

http://dlmf.nist.gov/37.12.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §37.12(ii), §37.12(ii), §37.12, Part 37.III and Ch.37

… ►

37.12.14 $$S_{\mathrm{\ell },m}^n(𝐱,t;\beta )=\underset{\gamma \to \mathrm{\infty }}{lim}{\gamma }^n{S}_{\mathrm{\ell },m}^n({\gamma }^{-1}𝐱,{\gamma }^{-1}t;\beta ,\gamma ).$$

ⓘ

Symbols:

$m$: nonnegative integer, $n$: nonnegative integer, $\mathrm{\ell }$: positive integer and $𝐱$: vector in $d$-dimensional Euclidean space

Proof sketch:

Use (18.7.22).

Permalink:

http://dlmf.nist.gov/37.12.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §37.12(iii), §37.12(iii), §37.12, Part 37.III and Ch.37

…

9: 37.19 Other Orthogonal Polynomials of $d$ Variables

… ►

37.19.1 $$T_{j}f(𝐱)=\frac{\partial f}{\partial x_j}+\sum _{𝐯\in R_{+}}{\kappa }_{𝐯}\frac{f(𝐱)-f(𝐱{\sigma }_{𝐯})}{⟨𝐱,𝐯⟩}{v}_j,$$

ⓘ

Symbols:

$\in$: element of, $⟨𝐮,𝐯⟩$: inner product over vectors, $\partial x$: partial differential of $x$ , $j$: nonnegative integer, $𝐱$: vector in $d$-dimensional Euclidean space and $x$: real variable

Sources:

Dunkl (1989, Definition 1.3); Dunkl and Xu (2014, Definition 6.4.2)

Permalink:

http://dlmf.nist.gov/37.19.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §37.19(i), §37.19, Part 37.III and Ch.37

… ►

37.19.4 $$w_{\kappa }(𝐱)=\prod _{𝐯\in R_{+}}{\left|⟨𝐱,𝐯⟩\right|}^{2{\kappa }_{𝐯}}.$$

ⓘ

Symbols:

$\in$: element of, $⟨𝐮,𝐯⟩$: inner product over vectors and $𝐱$: vector in $d$-dimensional Euclidean space

Sources:

Dunkl (1989, p. 168); Dunkl and Xu (2014, Definition 7.1.1)

Referenced by:

§37.19(ii)

Permalink:

http://dlmf.nist.gov/37.19.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §37.19(i), §37.19, Part 37.III and Ch.37

… ►

37.19.5 $$w_{\kappa }(𝐱)=\prod _{\mathrm{\ell }=1}^d{\left|x_{\mathrm{\ell }}\right|}^{2{\kappa }_{\mathrm{\ell }}},$$ ${\kappa }_{\mathrm{\ell }}\ge 0$.

ⓘ

Symbols:

$d$: positive integer, usually $\ge 2$ , $\mathrm{\ell }$: positive integer, $𝐱$: vector in $d$-dimensional Euclidean space and $x$: real variable

Source:

Dunkl and Xu (2014, Definition 7.5.1)

Referenced by:

§37.19(i), §37.19(iii)

Permalink:

http://dlmf.nist.gov/37.19.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §37.19(i), §37.19, Part 37.III and Ch.37

… ►

37.19.6 $$W_{\kappa ,\mu }(𝐱)=w_{\kappa }(𝐱){\left(1-{\Vert 𝐱\Vert }^2\right)}^{\mu -\frac{1}{2}}$$

ⓘ

Defines:

$W$: weight function in $d$-variables (locally)

Symbols:

$\Vert 𝐯\Vert$: vector norm ($l$²), $d$: positive integer, usually $\ge 2$ and $𝐱$: vector in $d$-dimensional Euclidean space

Source:

Dunkl and Xu (2014, Definition 8.1.1)

Permalink:

http://dlmf.nist.gov/37.19.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §37.19(ii), §37.19, Part 37.III and Ch.37

… ►

37.19.7 $$⟨f,g⟩={\int }_{{𝔹}^d}{\nabla }^{s}f(𝐱){\nabla }^{s}g(𝐱)d𝐱+\sum _{k=0}^{\lfloor \frac{s}{2}\rfloor -1}{\int }_{{𝕊}^{d-1}}{\mathrm{\Delta }}^{k}f(\xi ){\mathrm{\Delta }}^{k}g(\xi )d\sigma (\xi ),$$

ⓘ

Defines:

$⟨f,g⟩$: Sobolev inner product on the ball (locally)

Symbols:

$dx$: differential of $x$ , $\lfloor x\rfloor$: floor of $x$ , $\int$: integral, $k$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, ${𝕊}^{d-1}$: unit sphere of dimension $d-1$ and ${𝔹}^d$: unit ball

Permalink:

http://dlmf.nist.gov/37.19.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §37.19(iv), §37.19, Part 37.III and Ch.37

…

10: 3.2 Linear Algebra

… ►The $p$-norm of a vector $𝐱={[x_1,\mathrm{\dots },x_n]}^{\mathrm{T}}$ is given by …The Euclidean norm is the case $p=2$. … ►A normalized eigenvector has Euclidean norm 1; compare (3.2.13) with $p=2$. …