Dunkl operator

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1—10 of 17 matching pages

1: 18.38 Mathematical Applications

… ►

Dunkl Type Operators and Nonsymmetric Orthogonal Polynomials

►The Dunkl operator, introduced by Dunkl (1989), is an operator associated with reflection groups or root systems which has terms involving first order partial derivatives and reflection terms. Analogues of the original Dunkl operator (the rational case) were introduced by Heckman and Cherednik for the trigonometric case, and by Cherednik for the

q

-case. … ►In the one-variable case the Dunkl operator eigenvalue equation … …

2: 37.19 Other Orthogonal Polynomials of $d$ Variables

… ►For

1\leq j\leq d

, the Dunkl operator

T_{j}

is defined by …

3: 37.16 Orthogonal Polynomials on the Hyperoctant

… ►The spaces

\mathcal{V}_{n}^{\boldsymbol{{\alpha}}}(\mathbb{R}_{+}^{d})

are eigenspaces of a second order partial differential operator: ►

37.16.4

\sum_{\ell=1}^{d}\left(x_{\ell}{{D}_{x_{\ell}}}^{2}+(\alpha_{\ell}+1-x_{\ell})% {D}_{x_{\ell}}\right)u(x)=-n\,u(x),

u\in\mathcal{V}_{n}^{\boldsymbol{{\alpha}}}(\mathbb{R}_{+}^{d})

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\in$ : element of, $\mathbb{R}$ : real line, $n$ : nonnegative integer, $d$ : positive integer, usually $\geq 2$ , $\ell$ : positive integer, $\mathbb{R}_{+}^{d}$ : hyperoctant and $x$ : real variable
Proved:: Dunkl and Xu (2014, (5.1.9))(proved)
Referenced by:: §37.13(ii)
Permalink:: http://dlmf.nist.gov/37.16.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §37.16, Part 37.III and Ch.37

… ►

37.16.5

\prod_{\ell=1}^{d}L^{(\alpha_{\ell})}_{\nu_{\ell}}\left(x_{\ell}\right),

\nu_{1}+\cdots+\nu_{d}=n

ⓘ

Symbols:: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $n$ : nonnegative integer, $d$ : positive integer, usually $\geq 2$ , $\ell$ : positive integer and $x$ : real variable
Proved:: Dunkl and Xu (2014, (5.1.8))(proved)
Referenced by:: §37.16(iii)
Permalink:: http://dlmf.nist.gov/37.16.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §37.16(i), §37.16, Part 37.III and Ch.37

… ►

37.16.6

Q_{\boldsymbol{{\nu}},k}^{\boldsymbol{{\alpha}}}(\mathbf{x})=L^{(2k+|{% \boldsymbol{{\alpha}}}|+1)}_{n-k}\left(|{\mathbf{x}}|\right)\*|{\mathbf{x}}|^{% k}P_{\boldsymbol{{\nu}}}^{\boldsymbol{{\alpha}}}\left(\frac{x_{1}}{|{\mathbf{x% }}|},\ldots,\frac{x_{d-1}}{|{\mathbf{x}}|}\right),

0\leq k\leq n

\nu\in\mathbb{N}_{0}^{d-1}

|{\boldsymbol{{\nu}}}|=n-k

ⓘ

Symbols:: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $\in$ : element of, $\mathbb{N}$ : set of all positive integers, $k$ : nonnegative integer, $n$ : nonnegative integer, $P$ : polynomial of two variables, $Q$ : polynomial of two variables, $d$ : positive integer, usually $\geq 2$ , $\mathbf{x}$ : vector in $d$ -dimensional Euclidean space, $\mathbb{N}_{0}$ : set of all nonnegative integers and $x$ : real variable
Source:: Dunkl and Xu (2014, Proposition 8.4.7); There set $h_{\kappa}(x)=1$ .
Referenced by:: §37.16(iii)
Permalink:: http://dlmf.nist.gov/37.16.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §37.16(i), §37.16, Part 37.III and Ch.37

… ►

37.16.9

\lim_{\beta\to\infty}\beta^{k}P_{\boldsymbol{{\nu}}}^{\beta,\boldsymbol{{% \alpha}}}\left(1-\beta^{-1}|{\mathbf{x}}|,\beta^{-1}x_{1},\ldots,\beta^{-1}x_{% d-1}\right)=Q_{\boldsymbol{{\nu}},k}^{\boldsymbol{{\alpha}}}(\mathbf{x}),

\mathbf{x}=(x_{1},\ldots,x_{d})

|{\boldsymbol{{\nu}}}|=n

\nu_{1}=n-k

ⓘ

Symbols:: $k$ : nonnegative integer, $n$ : nonnegative integer, $P$ : polynomial of two variables, $Q$ : polynomial of two variables, $d$ : positive integer, usually $\geq 2$ , $\mathbf{x}$ : vector in $d$ -dimensional Euclidean space and $x$ : real variable
Source:: Dunkl and Xu (2014, Theorems 8.4.4 and 8.4.5)
Permalink:: http://dlmf.nist.gov/37.16.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §37.16(iii), §37.16, Part 37.III and Ch.37

4: 37.17 Hermite Polynomials on ${\mathbb{R}}^{d}$

… ►The spaces

\mathcal{V}_{n}({\mathbb{R}}^{d})

are eigenspaces of a second order partial differential operator, see (37.17.10). … ►

37.17.4

\sum_{|{\boldsymbol{{\nu}}}|=n}\frac{H_{\boldsymbol{{\nu}}}(\mathbf{x})}{% \boldsymbol{{\nu}}!}\mathbf{y}^{\boldsymbol{{\nu}}}=\frac{1}{n!\,{\pi}^{\frac{% 1}{2}}}\int_{-\infty}^{\infty}H_{n}\left(\left\langle\mathbf{x},\mathbf{y}% \right\rangle+t\sqrt{1-{\left\|{\mathbf{y}}\right\|}^{2}}\right){\mathrm{e}}^{% -t^{2}}\,\mathrm{d}t,

\left\|{\mathbf{y}}\right\|\leq 1

ⓘ

Symbols:: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$ : inner product over vectors, $\int$ : integral, $\left\|{\NVar{\mathbf{v}}}\right\|$ : vector norm ( $l$ ²), $n$ : nonnegative integer, $\mathbf{x}$ : vector in $d$ -dimensional Euclidean space and $\mathbf{y}$ : vector in $d$ -dimensional Euclidean space
Proved:: Dunkl and Xu (2014, p. 283)(proved)
Referenced by:: (37.17.5)
Permalink:: http://dlmf.nist.gov/37.17.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §37.17(i), §37.17(i), §37.17, Part 37.III and Ch.37

… ►The spaces

\mathcal{V}_{n}({\mathbb{R}}^{d})

are eigenspaces of a second order partial differential operator: ►

37.17.10

\left(\tfrac{1}{2}\Delta-\sum_{\ell=1}^{d}x_{\ell}{D}_{x_{\ell}}\right)u(x)=-n% \,u(x),

u\in\mathcal{V}_{n}({\mathbb{R}}^{d})

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\in$ : element of, $\mathbb{R}$ : real line, $n$ : nonnegative integer, $d$ : positive integer, usually $\geq 2$ , ${\mathbb{R}}^{d}$ : $d$ -dimensional Euclidean space, $\ell$ : positive integer 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, Part 37.III and Ch.37

►See (37.11.1) for the Laplace operator

\Delta

. …

5: Bibliography T

… ►

S. Tsujimoto, L. Vinet, and A. Zhedanov (2012) Dunkl shift operators and Bannai-Ito polynomials. Adv. Math. 229 (4), pp. 2123–2158.

ⓘ

External Links:: ISSN 0001-8708, Link, MathReview (Juan José Moreno-Balcázar), ZentralBlatt
Referenced by:: §18.28(xi).
Encodings:: BibTeX

…

6: 37.15 Orthogonal Polynomials on the Ball

… ►The spaces

\mathcal{V}_{n}^{\alpha}(\mathbb{B}^{d})

are eigenspaces of a second order partial differential operator, see (37.15.16). … ►

37.15.5

C_{\boldsymbol{{\nu}}}^{(\alpha+\frac{1}{2})}(\mathbf{x})=\prod_{j=1}^{d}\left% (1-\left(x_{1}^{2}+\cdots+x_{j-1}^{2}\right)\right)^{\frac{1}{2}\nu_{j}}C^{(% \alpha+\nu_{j+1}+\cdots+\nu_{d}+\frac{1}{2}(d-j+1))}_{\nu_{j}}\left(\frac{x_{j% }}{\sqrt{1-\left(x_{1}^{2}+\cdots+x_{j-1}^{2}\right)}}\right)

ⓘ

Symbols:: $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $j$ : nonnegative integer, $d$ : positive integer, usually $\geq 2$ , $\mathbf{x}$ : vector in $d$ -dimensional Euclidean space and $x$ : real variable
Proved:: Dunkl and Xu (2014, Proposition 5.2.2)(proved)
Referenced by:: §37.15(vii), (37.17.12), §37.17(v)
Permalink:: http://dlmf.nist.gov/37.15.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §37.15(ii), §37.15, Part 37.III and Ch.37

… ►See Dunkl and Xu (2014, Proposition 5.2.2) for the explicit value of

\langle{C_{\boldsymbol{{\nu}}}^{(\alpha+\frac{1}{2})}},{C_{\boldsymbol{{\nu}}}% ^{(\alpha+\frac{1}{2})}}\rangle_{\alpha}

. … ►See Dunkl and Xu (2014, §5.2.2) for expressions of of these biorthogonal polynomials in terms of Lauricella’s hypergeometric function

F_{B}

. … ►The spaces

\mathcal{V}_{n}^{\alpha}(\mathbb{B}^{d})

are eigenspaces of a second order partial differential operator: …

7: null

error generating summary

8: 37.4 Disk with Weight Function $\left(1-x^{2}-y^{2}\right)^{\alpha}$

… ►The spaces

\mathcal{V}_{n}^{\alpha}

are eigenspaces of a second order partial differential operator, see (37.4.28). … ►See Dunkl and Xu (2014, §3.3.4) for multi-term relations satisfied by the OPs (37.4.5). … ►See Dunkl and Xu (2014, Proposition 2.3.6). … ►For these results see Appell and Kampé de Fériet (1926, pp. 263, 269), Erdélyi et al. (1953b, §§12.5, 12.6) (with

s=2\alpha+1

) and Dunkl and Xu (2014, §2.3) (with

\mu=\alpha+\tfrac{1}{2}

). … ►The spaces

\mathcal{V}_{n}^{\alpha}

of OPs on

\mathbb{D}

are eigenspaces of a second order partial differential operator: …

9: 37.3 Triangular Region with Weight Function $x^{\alpha}y^{\beta}(1-x-y)^{\gamma}$

… ►The spaces

\mathcal{V}_{n}^{\alpha,\beta,\gamma}

are eigenspaces of a second order partial differential operator, see (37.3.14). … ►

37.3.3

P^{\alpha,\beta,\gamma}_{k,n}\left(x,y\right)=P^{(\beta+\gamma+2k+1,\alpha)}_{% n-k}\left(2x-1\right)\*(1-x)^{k}P^{(\gamma,\beta)}_{k}\left(\frac{2y}{1-x}-1% \right),

0\leq k\leq n

\alpha,\beta,\gamma>-1

ⓘ

Symbols:: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $k$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable, $y$ : real variable, $\alpha$ : real parameter ( $>-1$ ), $\beta$ : real parameter ( $>-1$ ) and $\gamma$ : real parameter ( $>-1$ )
Source:: Dunkl and Xu (2014, Proposition 2.4.1)
Referenced by:: (37.10.7), §37.10(iv), §37.14(ii), (37.3.17), (37.3.31), (37.3.32), (37.3.6), (37.3.7), §37.3(iv), §37.3(v), §37.4(vii), §37.5(iii), §37.7(ii)
Permalink:: http://dlmf.nist.gov/37.3.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §37.3(ii), §37.3, Part 37.II and Ch.37

… ►

37.3.4

\langle{P^{\alpha,\beta,\gamma}_{k,n}},{P^{\alpha,\beta,\gamma}_{j,m}}\rangle_% {\alpha,\beta,\gamma}=h_{k,n}^{\alpha,\beta,\gamma}\delta_{n,m}\delta_{k,j},

\alpha,\beta,\gamma>-1

ⓘ

Defines:: $h_{k,n}^{\alpha,\beta,\gamma}$ : positive constant (locally)
Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $j$ : nonnegative integer, $k$ : nonnegative integer, $m$ : nonnegative integer, $n$ : nonnegative integer, $\langle{f},{g}\rangle_{\alpha,\beta,\gamma}$ : inner product on triangular region, $\alpha$ : real parameter ( $>-1$ ), $\beta$ : real parameter ( $>-1$ ) and $\gamma$ : real parameter ( $>-1$ )
Proved:: Dunkl and Xu (2014, Proposition 2.4.1)(proved)
Referenced by:: (37.3.32)
Permalink:: http://dlmf.nist.gov/37.3.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §37.3(ii), §37.3, Part 37.II and Ch.37

… ►The coefficients in these expressions are

{{}_{4}F_{3}}

hypergeometric functions that can be written in terms of Racah polynomials, see (Dunkl, 1984, Theorem 1.7) and (Iliev and Xu, 2017, Proposition 4.2). … ►The spaces

\mathcal{V}_{n}^{\alpha,\beta,\gamma}

of OPs on

\triangle

are eigenspaces of a second order partial differential operator (PDO): …

10: 18.28 Askey–Wilson Class

… ►In Tsujimoto et al. (2012) an extension of the Bannai–Ito polynomials occurs as eigenfunctions of a Dunkl type operator. …

Dunkl operator

1—10 of 17 matching pages

1: 18.38 Mathematical Applications

Dunkl Type Operators and Nonsymmetric Orthogonal Polynomials

2: 37.19 Other Orthogonal Polynomials of d Variables

3: 37.16 Orthogonal Polynomials on the Hyperoctant

4: 37.17 Hermite Polynomials on ℝ d

5: Bibliography T

6: 37.15 Orthogonal Polynomials on the Ball

7: null

8: 37.4 Disk with Weight Function ( 1 − x 2 − y 2 ) α

9: 37.3 Triangular Region with Weight Function x α ⁢ y β ⁢ ( 1 − x − y ) γ

10: 18.28 Askey–Wilson Class

2: 37.19 Other Orthogonal Polynomials of $d$ Variables

4: 37.17 Hermite Polynomials on ${\mathbb{R}}^{d}$

8: 37.4 Disk with Weight Function $\left(1-x^{2}-y^{2}\right)^{\alpha}$

9: 37.3 Triangular Region with Weight Function $x^{\alpha}y^{\beta}(1-x-y)^{\gamma}$