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1: 37.1 Notation

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$d$	positive integer, usually $\ge 2$.
…
${ℝ}^d$	$d$-dimensional Euclidean space.
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$𝐱,𝐲$	$(x_1,\mathrm{\dots },x_d),(y_1,\mathrm{\dots },y_d)\in {ℝ}^d$.
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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, $𝐱$: vector in $d$-dimensional Euclidean space, $f$: integrable function on ${ℝ}^d$ and $g$: integrable function on ${ℝ}^d$

Referenced by:

§37.17

Permalink:

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

Encodings:

pMML, png, TeX

See also:

Annotations for §37.17 and Ch.37

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

ⓘ

Symbols:

$H_n\left(x\right)$: Hermite polynomial, $H_{𝝂}\left(𝐱\right)$: Hermite product polynomial, $!$: factorial (as in $n!$), $!$: factorial for multi-indices, $⟨𝐮,𝐯⟩$: inner product over vectors, $\Vert 𝐯\Vert$: vector norm ($l$²), $n$: nonnegative integer, $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space, $|𝐱|$: $l1$ norm and $𝝂$: multi-index of $d$-dimensions

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 and Ch.37

►

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

ⓘ

Symbols:

$H_{𝝂}\left(𝐱\right)$: Hermite product polynomial, $\in$: element of, $\exp z$: exponential function, $!$: factorial for multi-indices, $⟨𝐮,𝐯⟩$: 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, $𝝂$: multi-index of $d$-dimensions 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 and Ch.37

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37.17.10 $$\left(\frac{1}{2}\mathrm{\Delta }-\sum _{\mathrm{\ell }=1}^d{x}_{\mathrm{\ell }}{D}_{x_{\mathrm{\ell }}}\right)u(𝐱)=-nu(𝐱),$$ $u\in {𝒱}_n\left({ℝ}^d\right)$.

ⓘ

Defines:

$\mathrm{\Delta }$: Laplacian (locally)

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\in$: element of, $ℝ$: real line, ${𝒱}_n\left({ℝ}^d\right)$: vector space of $d$-variable orthogonal polynomials with respect to the Gaussian weight, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , ${ℝ}^d$: $d$-dimensional Euclidean space, $\mathrm{\ell }$: positive integer, $𝐱$: vector in $d$-dimensional Euclidean space, $u$: orthogonal polynomial in $d$-variables with respect to the Gaussian weight and $x$: real variable

Proved:

Dunkl and Xu (2014, (5.1.7))(proved)

Referenced by:

§37.13(ii), §37.17

Permalink:

http://dlmf.nist.gov/37.17.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §37.17(iii), §37.17 and Ch.37

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37.17.19 $$\sum _{𝝂\in {\mathbb{N}}_0^d}{H}_{𝝂}(𝐱;𝐀)\frac{{𝐲}^{𝝂}}{𝝂!}={\mathrm{e}}^{2⟨𝐀𝐲,𝐱⟩-⟨𝐀𝐲,𝐲⟩}.$$

ⓘ

Symbols:

$H_{𝝂}(𝐱;𝐀)$: Hermite biorthogonal base $H$ , $\in$: element of, $\mathrm{e}$: base of natural logarithm, $!$: factorial for multi-indices, $⟨𝐮,𝐯⟩$: inner product over vectors, $\mathbb{N}$: set of all positive integers, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space, $𝝂$: multi-index of $d$-dimensions, ${\mathbb{N}}_0$: set of all nonnegative integers and $𝐀$: positive definite $d\times d$ matrix

Source:

Erdélyi et al. (1953b, (17) in §12.8)

Permalink:

http://dlmf.nist.gov/37.17.E19

Encodings:

pMML, png, TeX

See also:

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

…

3: 37.13 General Orthogonal Polynomials of $d$ Variables

… ►Let $W$ be a nonnegative weight function on an open set $\mathrm{\Omega }$ in ${ℝ}^d$ such that the integral ${\int }_{\mathrm{\Omega }}P(𝐱)W(𝐱)d𝐱$ is well-defined and absolutely convergent for all polynomials $P$, and such that ${\int }_{\mathrm{\Omega }}W(𝐱)d𝐱>0$. … ►

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 $\ge 2$ , $\mathrm{\Omega }$: open set in $d$-dimensional Euclidean space, $W$: nonnegative weight function in $d$-variables on open set $\mathrm{\Omega }$ , $f$: integrable function on $d$-dimensional Euclidean space, $g$: integrable function on $d$-dimensional Euclidean space and $𝐱$: $d$-dimensional vector of real numbers

Referenced by:

§37.13

Permalink:

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

Encodings:

pMML, png, TeX

See also:

Annotations for §37.13 and Ch.37

… ►

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

ⓘ

Defines:

$Q$: polynomial of $d$-variables (locally) and ${𝐑}_n$: reproducing kernel on ${𝒱}_n^d$ (locally)

Symbols:

$dx$: differential of $x$ , $\in$: element of, $\int$: integral, $ℝ$: real line, ${𝒱}_n^d$: vector space of orthogonal orthogonal polynomials in $d$-variables, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , ${ℝ}^d$: $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space, $\mathrm{\Omega }$: open set in $d$-dimensional Euclidean space, $W$: nonnegative weight function in $d$-variables on open set $\mathrm{\Omega }$ and $𝐱$: $d$-dimensional vector of real numbers

Permalink:

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

Encodings:

pMML, png, TeX

See also:

Annotations for §37.13, §37.13 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, ${ℝ}^d$: $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space, $|𝐱|$: $l1$ norm, $𝝂$: multi-index of $d$-dimensions, ${𝐑}_n$: reproducing kernel on ${𝒱}_n^d$ , $W$: nonnegative weight function in $d$-variables on open set $\mathrm{\Omega }$ , $P$: polynomial of $d$-variables, ${⟨f,g⟩}_W$: inner product on $d$-dimensional Euclidean space and $𝐱$: $d$-dimensional vector of real numbers

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 and Ch.37

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

ⓘ

Defines:

${𝐏}_z$: Poisson kernel on ${𝒱}_n^d$ (locally)

Symbols:

$|z|$: absolute value (modulus) for complex numbers, ${𝒱}_n^d$: vector space of orthogonal orthogonal polynomials in $d$-variables, $n$: nonnegative integer, $z$: complex number, $d$: positive integer, usually $\ge 2$ , $𝐲$: vector in $d$-dimensional Euclidean space, ${𝐑}_n$: reproducing kernel on ${𝒱}_n^d$ and $𝐱$: $d$-dimensional vector of real numbers

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 and Ch.37

…

4: 37.16 Orthogonal Polynomials on the Hyperoctant

… ►

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

ⓘ

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 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 on the hyperoctant (locally)

Symbols:

$\mathrm{e}$: base of natural logarithm, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, $𝜶$: vector $=({\alpha }_1,\mathrm{\dots },{\alpha }_{d+1})$ and $|𝐱|$: $l1$ norm

Permalink:

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

Encodings:

pMML, png, TeX

See also:

Annotations for §37.16 and Ch.37

… ►

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, $𝜶$: vector $=({\alpha }_1,\mathrm{\dots },{\alpha }_{d+1})$ , ${ℝ}_{+}^d$: hyperoctant and $W$: weight function in $d$-variables on the hyperoctant

Referenced by:

§37.16

Permalink:

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

Encodings:

pMML, png, TeX

See also:

Annotations for §37.16 and Ch.37

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

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $|z|$: absolute value (modulus) for complex numbers, $\in$: element of, $\exp z$: exponential function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $ℝ$: real line, $n$: nonnegative integer, $z$: complex number, $d$: positive integer, usually $\ge 2$ , $\mathrm{\ell }$: positive integer, $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space, $𝜶$: vector $=({\alpha }_1,\mathrm{\dots },{\alpha }_{d+1})$ , $|𝐱|$: $l1$ norm, ${𝐑}_n$: reproducing kernel on ${𝒱}_n^d$ , ${𝐏}_z$: Poisson kernel on ${𝒱}_n^d$ , ${ℝ}_{+}^d$: hyperoctant, $x$: 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 and Ch.37

… ►

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

ⓘ

Symbols:

$P_{𝝂}^{𝜶}$: Jacobi polynomial on the simplex, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, $𝜶$: vector $=({\alpha }_1,\mathrm{\dots },{\alpha }_{d+1})$ , $|𝐱|$: $l1$ norm, $𝝂$: multi-index of $d$-dimensions 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 and Ch.37

…

5: 37.18 Orthogonal Polynomials on Quadratic Domains

… ►

37.18.1 $${𝕍}^{d+1}=\{(𝐱,t)\mid \Vert 𝐱\Vert \le \left|\varphi (t)\right|,t\in [a,b],𝐱\in {ℝ}^d\},$$

ⓘ

Defines:

${𝕍}^{d+1}$: quadratic domain (locally)

Symbols:

$[a,b]$: closed interval, $\in$: element of, $(a,b)$: open interval, $ℝ$: real line, $\Vert 𝐯\Vert$: vector norm ($l$²), ${ℝ}^d$: $d$-dimensional Euclidean space, $𝐱$: vector in $d$-dimensional Euclidean space, $t$: real variable in $[a,b]$ , $a$: variable in $ℝ\cup \{-\mathrm{\infty }\}$ , $b$: variable in $ℝ\cup \{+\mathrm{\infty }\}$ and $\varphi$: function on $[a,b]$

Permalink:

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

Encodings:

pMML, png, TeX

See also:

Annotations for §37.18 and Ch.37

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37.18.2 $$W(𝐱,t)=w_1(t)w_2\left(\frac{𝐱}{\varphi (t)}\right).$$

ⓘ

Defines:

$W$: weight function in $(d+1)$-variables on the quadratic domain (locally)

Symbols:

$d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, $t$: real variable in $[a,b]$ , $\varphi$: function on $[a,b]$ , $w_1$: weight function and $w_2$: weight function on ${ℝ}^d$

Referenced by:

§37.18(ii), §37.18(iii)

Permalink:

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

Encodings:

pMML, png, TeX

See also:

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

… ►

37.18.7 ,

ⓘ

Defines:

: weight function on ${ℝ}^{d+1}$ (locally)

Symbols:

$ℝ$: 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, : real constant $(>-\frac{1}{2})$ , $\beta$: real constant, : real constant $(>-1)$ and $t$: real constant in $[0,1]$

Permalink:

http://dlmf.nist.gov/37.18.E7

Encodings:

pMML, png, TeX

See also:

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

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37.18.11 $$\zeta (𝐱,t,𝐲,s;u,𝐯)=v_1\sqrt{\frac{1}{2}\left(ts+⟨𝐱,𝐲⟩+u\sqrt{t^2-{\Vert 𝐱\Vert }^2}\sqrt{s^2-{\Vert 𝐲\Vert }^2}\right)}+v_2\sqrt{1-t}\sqrt{1-s};$$

ⓘ

Defines:

$\zeta$: function (locally)

Symbols:

$⟨𝐮,𝐯⟩$: inner product over vectors, $\Vert 𝐯\Vert$: vector norm ($l$²), $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space, $t$: real constant in $[0,1]$ , $u$: OP in $(d+1)$-variables on the bounded cone and $𝐯$: vector $(v_1,v_2)$

Permalink:

http://dlmf.nist.gov/37.18.E11

Encodings:

pMML, png, TeX

See also:

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

… ►

37.18.12 ,

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $ℝ$: 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, $t$: real variable in $[a,b]$ , $W$: weight function in $d$-variables on ball, $\mu$: real constant $(\ge -\frac{1}{2})$ and : real constant $(>-1)$

Permalink:

http://dlmf.nist.gov/37.18.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §37.18(iii), §37.18 and Ch.37

…

6: 37.11 Spherical Harmonics

… ►

37.11.1 $$\Delta f=\sum _{j=1}^d\frac{{\partial }^{2}f}{{\partial x_j}^2}.$$

ⓘ

Defines:

$\Delta$: Laplacian on Euclidean space

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $j$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $f$: smooth function of $d$-variables and $x$: real variable

Referenced by:

§37.17(iii)

Permalink:

http://dlmf.nist.gov/37.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §37.11 and Ch.37

… ►The spherical part of the Laplacian $\Delta$ on ${ℝ}^d$ is a differential operator ${\mathrm{\Delta }}_0={\mathrm{\Delta }}_{0,d}$ on ${𝕊}^{d-1}$ defined by ►

37.11.20 $${\mathrm{\Delta }}_{0}f(𝝃)=\Delta F(𝝃).$$

ⓘ

Symbols:

$\Delta$: Laplacian on Euclidean space, ${\mathrm{\Delta }}_0$: spherical part of the Laplacian on Euclidean space, $f$: smooth function of $d$-variables, $𝝃$: $d$-dimensional vector restricted to ${𝕊}^{d-1}$ and $F$: zonal polynomial restricted to ${𝕊}^{d-1}$

Permalink:

http://dlmf.nist.gov/37.11.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §37.11(ii), §37.11 and Ch.37

… ►

37.11.21 $$\Delta F(𝐱)=\left(\frac{{\partial }^2}{{\partial r}^2}+\frac{d-1}{r}\frac{\partial }{\partial r}+\frac{1}{r^2}{\mathrm{\Delta }}_0\right)F(𝐱),$$

ⓘ

Symbols:

$\Delta$: Laplacian on Euclidean space, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , ${\mathrm{\Delta }}_0$: spherical part of the Laplacian on Euclidean space, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, $F$: zonal polynomial restricted to ${𝕊}^{d-1}$ and $r$: non-negative real parameter

Permalink:

http://dlmf.nist.gov/37.11.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §37.11(ii), §37.11 and Ch.37

… ►The Fourier transform of a function $f$ on ${ℝ}^d$ is a function $ℱ\left(f\right)$ on ${ℝ}^d$ which is defined by the Fourier integral (1.16.29). …

7: 37.15 Orthogonal Polynomials on the Ball

… ►

37.15.1

ⓘ

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 and Ch.37

… ►

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

ⓘ

Defines:

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

Symbols:

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

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 and Ch.37

… ►

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

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\in$: element of, $!$: factorial for multi-indices, $⟨𝐮,𝐯⟩$: inner product over vectors, $V_{𝝂}^{(𝜶)}\left(𝐱\right)$: monic basis element on the ball, $\mathbb{N}$: 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, $|𝐱|$: $l1$ norm, $𝝂$: multi-index of $d$-dimensions, ${\mathbb{N}}_0$: set of all nonnegative integers and $\alpha$: real variable

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 and Ch.37

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37.15.14 $$C_n^{(\alpha +\frac{1}{2}d)}(⟨𝐱,𝐲⟩)=\sum _{|𝝂|=n}\frac{2^n{\left(\alpha +\frac{1}{2}d\right)}_n}{𝝂!}{𝐲}^{𝝂}{V}_{𝝂}^{(\alpha +\frac{1}{2})}\left(𝐱\right),$$ $𝐲\in {𝕊}^{d-1}$.

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\in$: element of, $!$: factorial for multi-indices, $⟨𝐮,𝐯⟩$: inner product over vectors, $V_{𝝂}^{(𝜶)}\left(𝐱\right)$: monic basis element on the ball, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space, $|𝐱|$: $l1$ norm, $𝝂$: multi-index of $d$-dimensions, ${𝕊}^{d-1}$: unit sphere of dimension $d-1$ and $\alpha$: real variable

Proof sketch:

Use (18.12.4) and (37.15.13).

Permalink:

http://dlmf.nist.gov/37.15.E14

Encodings:

pMML, png, TeX

See also:

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

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37.15.15 $${\left({(1-⟨𝐱,𝐲⟩)}^2+{\Vert 𝐲\Vert }^2\left(1-{\Vert 𝐱\Vert }^2\right)\right)}^{-\alpha -\frac{1}{2}}=\sum _{𝝂\in {\mathbb{N}}_0^d}\frac{{(-1)}^{|𝝂|}{\left(2\alpha +1\right)}_{|𝝂|}}{2^{|𝝂|}{\left(\alpha +1\right)}_{|𝝂|}𝝂!}{U}_{𝝂}^{(\alpha +\frac{1}{2})}\left(𝐱\right){𝐲}^{𝝂},$$ $𝐲\in {ℝ}^d$, .

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $U_{𝝂}^{(𝜶)}\left(𝐱\right)$: co-monic basis element on the ball, $\in$: element of, $!$: factorial for multi-indices, $⟨𝐮,𝐯⟩$: 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, $|𝐱|$: $l1$ norm, $𝝂$: multi-index of $d$-dimensions, ${\mathbb{N}}_0$: set of all nonnegative integers and $\alpha$: real variable

Proved:

Dunkl and Xu (2014, Proposition 5.2.3)(proved)

Permalink:

http://dlmf.nist.gov/37.15.E15

Encodings:

pMML, png, TeX

See also:

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

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8: 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, $|𝐱|$: $l1$ norm 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 and Ch.37

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37.14.2 , , $𝐱\in {\mathrm{△}}^d$,

ⓘ

Defines:

$W_{𝜶}$: weight function on $d$-variables on the simplex (locally) and : 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, $𝜶$: vector $=({\alpha }_1,\mathrm{\dots },{\alpha }_{d+1})$ , $|𝐱|$: $l1$ norm, ${\mathrm{△}}^d$: 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 and Ch.37

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37.14.3

ⓘ

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 $\ge 2$ , $𝟏$: multi-dimensional vector with all components being unity, $\mathrm{\ell }$: positive integer, $𝐱$: vector in $d$-dimensional Euclidean space, $𝜶$: vector $=({\alpha }_1,\mathrm{\dots },{\alpha }_{d+1})$ , $|𝐱|$: $l1$ norm, ${\mathrm{△}}^d$: simplex, $g$: integrable function on $(-1,1)$ , $W_{𝜶}$: weight function on $d$-variables on the simplex, : real parameter ($>-1$) and $f$: $d$-variable integrable function on the simplex ${\mathrm{△}}^d$

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 and Ch.37

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37.14.7

ⓘ

Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $P_{𝝂}^{𝜶}$: Jacobi polynomial on the simplex, $j$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $\mathrm{\ell }$: positive integer, $𝐱$: vector in $d$-dimensional Euclidean space, $𝜶$: vector $=({\alpha }_1,\mathrm{\dots },{\alpha }_{d+1})$ , $𝝂$: multi-index of $d$-dimensions, : real parameter ($>-1$) and $x$: real variable

Source:

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

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 and Ch.37

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37.14.14

ⓘ

Defines:

$\zeta$: function on ${ℝ}^d\times {ℝ}^d\times {ℝ}^{d+1}$ (locally)

Symbols:

$ℝ$: real line, $d$: 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, $|𝐱|$: $l1$ norm, : vector such that , : real number with , $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/37.14.E14

Encodings:

pMML, png, TeX

See also:

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

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9: 37.19 Other Orthogonal Polynomials of $d$ Variables

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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, $𝐱{\sigma }_{𝐯}$: reflection, $f$: $(d-1)$-variable integrable function on ${𝕊}^{d-1}$ , $R_{+}$: set of positive roots, $T_j$: Dunkl operator, $𝐯$: nonnegative function on $R_{+}$ , ${\kappa }_{𝐯}$: something, $v_{\mathrm{\ell }}$: $\mathrm{\ell }$th component of $𝐯$ 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 and Ch.37

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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, $𝐱$: vector in $d$-dimensional Euclidean space, $w_{\kappa }$: weight function, $R_{+}$: set of positive roots, $𝐯$: nonnegative function on $R_{+}$ and ${\kappa }_{𝐯}$: something

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 and Ch.37

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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, $w_{\kappa }$: weight function, ${\kappa }_{\mathrm{\ell }}$: something 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 and Ch.37

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

ⓘ

Symbols:

$\Vert 𝐯\Vert$: vector norm ($l$²), $𝐱$: vector in $d$-dimensional Euclidean space, $W$: weight function in $d$-variables on ball, $\kappa$: real constant and $\mu$: real constant

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 and Ch.37

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

ⓘ

Defines:

$⟨f,g⟩$: Sobolev inner product on the ball (locally), $𝝃$: $d$-dimensional vector restricted to ${𝕊}^{d-1}$ (locally), $\mathrm{\nabla }$: gradient (locally) and $\mathrm{\Delta }$: Laplacian (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 $d-1$ , ${𝔹}^d$: unit ball, $f$: $d$-variable integrable function on ${𝔹}^d$ and $g$: $d$-variable integrable function on ${𝔹}^d$

Permalink:

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

Encodings:

pMML, png, TeX

See also:

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

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10: 37.12 Orthogonal Polynomials on Quadratic Surfaces

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37.12.1 $${𝕍}_0^{d+1}=\{(𝐱,t)\mid \Vert 𝐱\Vert =\left|\varphi (t)\right|,t\in [a,b],𝐱\in {ℝ}^d\},$$

ⓘ

Defines:

${𝕍}_0^{d+1}$: quadratic surface (locally) and $t$: real variable in $[a,b]$ (locally)

Symbols:

$[a,b]$: closed interval, $\in$: element of, $(a,b)$: open interval, $ℝ$: real line, $\Vert 𝐯\Vert$: vector norm ($l$²), ${ℝ}^d$: $d$-dimensional Euclidean space, $𝐱$: vector in $d$-dimensional Euclidean space, $a$: variable in $ℝ\cup \{-\mathrm{\infty }\}$ , $b$: variable in $ℝ\cup \{+\mathrm{\infty }\}$ and $\varphi$: function on $[a,b]$

Permalink:

http://dlmf.nist.gov/37.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §37.12 and Ch.37

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37.12.2 $$⟨f,g⟩=c_w{\int }_{{𝕍}_0^{d+1}}f(𝐱,t)g(𝐱,t)w(t)d\sigma (𝐱,t)$$

ⓘ

Defines:

$c_w$: normalization constant such that $⟨1,1⟩=1$ (locally)

Symbols:

$dx$: differential of $x$ , $\int$: integral, $(a,b)$: open interval, $𝐱$: vector in $d$-dimensional Euclidean space, ${𝕍}_0^{d+1}$: quadratic surface, $f$: integrable function on ${𝕍}_0^{d+1}$ , $g$: integrable function on ${𝕍}_0^{d+1}$ , $⟨f,g⟩$: inner product on quadratic surface, $t$: real variable in $[a,b]$ and $w$: weight function on $(a,b)$

Permalink:

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

Encodings:

pMML, png, TeX

See also:

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

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37.12.9 $$S_{\mathrm{\ell },m}^n(𝐱,t;\beta ,\gamma )=R_{n-m}^{(\gamma ,\beta +2m+d-1)}\left(2t-1\right)t^m{Y}_{\mathrm{\ell }}^m\left(\frac{𝐱}{t}\right).$$

ⓘ

Defines:

$S_{\mathrm{\ell },m}^n$: orthogonal basis on quadratic surface Jacobi

Symbols:

$R_n^{(\alpha ,\beta )}\left(x\right)$: standardized Jacobi polynomial, $n$: nonnegative integer, $m$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $\mathrm{\ell }$: positive integer, $𝐱$: vector in $d$-dimensional Euclidean space, $Y_{\mathrm{\ell }}^m$: orthonormal basis element for spherical harmonics in ${\mathscr{H}}_m^{0,d}$ , $\beta$: real parameter, $t$: real variable in $[a,b]$ and $\gamma$: real parameter

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.E9

Encodings:

pMML, png, TeX

See also:

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

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37.12.11 $${𝐑}_n((𝐱,t),(𝐲,s);w_{-1,\gamma })=\frac{{\displaystyle {\int }_{{[-1,1]}^2}}{Z}_{2n}^{(\gamma +d)}\left(\zeta (𝐱,t,𝐲,s;𝐯)\right){(1-v_1^2)}^{\frac{d-4}{2}}{(1-v_2^2)}^{\gamma -\frac{1}{2}}{dv}_1{dv}_2}{{\displaystyle {\int }_{{[-1,1]}^2}}{(1-v_1^2)}^{\frac{d-4}{2}}{(1-v_2^2)}^{\gamma -\frac{1}{2}}{dv}_1{dv}_2},$$ $d>2,\gamma >-\frac{1}{2}$,

ⓘ

Symbols:

$[a,b]$: closed interval, $dx$: differential of $x$ , $\int$: integral, $(a,b)$: open interval, $Z_n^{(\lambda )}\left(x\right)$: zonal polynomial, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space, $s$: real variable in $[a,b]$ , ${𝐑}_n$: reproducing kernel on ${𝒱}_n({𝕍}_0^{d+1},w)$ , $t$: real variable in $[a,b]$ , $\gamma$: real parameter, $w_{\beta ,\gamma }$: weight function associated with Jacobi weight, $zeta$: real function and $𝐯$: vector $(v_1,v_2)$

Proved:

Xu (2020, Theorem 8.2)(proved)

Referenced by:

§37.12(ii)

Permalink:

http://dlmf.nist.gov/37.12.E11

Encodings:

pMML, png, TeX

See also:

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

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37.12.12 $$\zeta (𝐱,t,𝐲,s;𝐯)=v_1\sqrt{\frac{ts+⟨𝐱,𝐲⟩}{2}}+v_2\sqrt{1-t}\sqrt{1-s},$$

ⓘ

Defines:

$zeta$: real function (locally)

Symbols:

$⟨𝐮,𝐯⟩$: inner product over vectors, $z$: complex number, $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space, $s$: real variable in $[a,b]$ , $t$: real variable in $[a,b]$ , $𝐯$: vector $(v_1,v_2)$ and $a$: variable in $ℝ\cup \{-\mathrm{\infty }\}$

Permalink:

http://dlmf.nist.gov/37.12.E12

Encodings:

pMML, png, TeX

See also:

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

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