Dunkl type operator

SearchAdvancedHelp

(0.001 seconds)

11—20 of 164 matching pages

11: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►Of importance for special functions are infinite determinants of Hill’s type. These have the property that the double series …Hill-type determinants always converge. … ►

§1.3(iv) Matrices as Linear Operators

►

Linear Operators in Finite Dimensional Vector Spaces

…

12: Bibliography K

… ►

Orthonormal polynomials with generalized Freud-type weights. J. Approx. Theory 121 (1), pp. 13–53.

ⓘ

External Links:

ISSN 0021-9045, Document, Link, MathReview (Walter Van Assche)

Referenced by:

§18.32.

Encodings:

BibTeX

… ►

Algorithm 763: INTERVAL_ARITHMETIC: A Fortran 90 module for an interval data type. ACM Trans. Math. Software 22 (4), pp. 385–392.

ⓘ

External Links:

ISSN 0098-3500, Document, GAMS (module 763; package TOMS), ZentralBlatt

Referenced by:

2nd item.

Encodings:

BibTeX

… ►

On differential equations for Sobolev-type Laguerre polynomials. Trans. Amer. Math. Soc. 350 (1), pp. 347–393.

ⓘ

External Links:

ISSN 0002-9947, FullText, MathReview (Hans W. Volkmer), ZentralBlatt

Referenced by:

§18.36(ii).

Encodings:

BibTeX

… ►

Jacobi polynomials, Bernstein-type inequalities and dispersion estimates for the discrete Laguerre operator. Adv. Math. 333, pp. 796–821.

ⓘ

External Links:

ISSN 0001-8708, Document, Link, MathReview (Martin Axel Hallnäs)

Referenced by:

§18.14(i).

Encodings:

BibTeX

… ►

Algebras of Virasoro type, the energy-momentum tensor, and operator expansions on Riemann surfaces. Funktsional. Anal. i Prilozhen. 23 (1), pp. 24–40 (Russian).

ⓘ

Notes:

In Russian. English translation: Funct. Anal. Appl. 23(1989), no. 1, pp. 19–33

External Links:

ISSN 0374-1990, Document, MathReview (Fedor Malikov), ZentralBlatt

Referenced by:

§20.12(ii).

Encodings:

BibTeX

…

13: 18.27 $q$-Hahn Class

… ►The $q$-Hahn class OP’s comprise systems of OP’s $\{p_n(x)\}$, $n=0,1,\mathrm{\dots },N$, or $n=0,1,2,\mathrm{\dots }$, that are eigenfunctions of a second order $q$-difference operator. …In the $q$-Hahn class OP’s the role of the operator $d/dx$ in the Jacobi, Laguerre, and Hermite cases is played by the $q$-derivative ${𝒟}_q$, as defined in (17.2.41). … ►

18.27.6 $$P_n^{(\alpha ,\beta )}(x;c,d;q)=\frac{c^n{q}^{-(\alpha +1)n}{(q^{\alpha +1},-q^{\alpha +1}{c}^{-1}d;q)}_n}{{(q,-q;q)}_n}{P}_n(q^{\alpha +1}{c}^{-1}x;q^{\alpha },q^{\beta },-q^{\alpha }{c}^{-1}d;q),$$

ⓘ

Defines:

$P_n^{(\alpha ,\beta )}(x;c,d;q)$: big $q$-Jacobi polynomial and $P_n^{(\alpha ,\beta )}(x;c,d;q)$ (locally)

Symbols:

$(a,b)$: open interval, $P_n(x;a,b,c;q)$: big $q$-Jacobi polynomial, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: real variable, $n$: nonnegative integer and $x$: real variable

Source:

Andrews and Askey (1985, (3.28))

Referenced by:

(18.27.12_5), §18.27(iii), Erratum (V1.0.17) for Equation (18.27.6)

Permalink:

http://dlmf.nist.gov/18.27.E6

Encodings:

pMML, png, TeX

Errata (effective with 1.0.17):

Originally the first argument of the big $q$-Jacobi polynomial on the right-hand side was written incorrectly as $q^{\alpha +1}{c}^{-1}dx$.

Reported 2017-09-27 by Tom Koornwinder

See also:

Annotations for §18.27(iii), §18.27 and Ch.18

… ►

18.27.10 $$p_n(x)=P_n^{(\alpha ,\beta )}(x;c,d;q)$$

ⓘ

Symbols:

$P_n^{(\alpha ,\beta )}(x;c,d;q)$: big $q$-Jacobi polynomial, $p_n(x)$: polynomial of degree $n$ , $q$: real variable, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.27.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §18.27(iii), §18.27 and Ch.18

… ►

18.27.12_5 $$\underset{q\to 1}{lim}{P}_n^{(\alpha ,\beta )}(x;c,d;q)={\left(\frac{c+d}{2}\right)}^n{P}_n^{(\alpha ,\beta )}\left(\frac{2x-c+d}{c+d}\right).$$

ⓘ

Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $P_n^{(\alpha ,\beta )}(x;c,d;q)$: big $q$-Jacobi polynomial, $q$: real variable, $n$: nonnegative integer and $x$: real variable

Source:

Andrews and Askey (1985, (3.33)); for $c=d=1$

Proof sketch:

On the left substitute first (18.27.6) and next (18.27.5). Then, after taking the limit, a ${}_2{}^{}F_1^{}$ hypergeometric function occurs. Finally use (18.5.7).

Referenced by:

§18.27(iii), §18.27(iii), Erratum (V1.2.0) Section 18.27

Permalink:

http://dlmf.nist.gov/18.27.E12_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.27(iii), §18.27(iii), §18.27 and Ch.18

…

14: Publications

… ►

The DLMF Project: A New Initiative in Classical Special Functions, in Special Functions—Proceedings of the International Workshop, Hong Kong, June 21-25, 1999 (C. Dunkl, M. Ismail, R. Wong, eds.), World Scientific, pp. 207–220.

…

15: 27.19 Methods of Computation: Factorization

… ►Techniques for factorization of integers fall into three general classes: Deterministic algorithms, Type I probabilistic algorithms whose expected running time depends on the size of the smallest prime factor, and Type II probabilistic algorithms whose expected running time depends on the size of the number to be factored. … ►Type I probabilistic algorithms include the Brent–Pollard rho algorithm (also called Monte Carlo method), the Pollard $p-1$ algorithm, and the Elliptic Curve Method (ecm). … ►Type II probabilistic algorithms for factoring $n$ rely on finding a pseudo-random pair of integers $(x,y)$ that satisfy $x^2\equiv y^2(modn)$. …As of January 2009 the snfs holds the record for the largest integer that has been factored by a Type II probabilistic algorithm, a 307-digit composite integer. …The largest composite numbers that have been factored by other Type II probabilistic algorithms are a 63-digit integer by cfrac, a 135-digit integer by mpqs, and a 182-digit integer by gnfs. …

16: 31.10 Integral Equations and Representations

… ►

§31.10(i) Type I

… ►

31.10.3 $$({𝒟}_z-{𝒟}_t)𝒦=0,$$

ⓘ

Defines:

$𝒦(z,t)$: kernel (locally)

Symbols:

$z$: complex variable and ${𝒟}_z$: Heun’s operator

Permalink:

http://dlmf.nist.gov/31.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §31.10(i), §31.10 and Ch.31

►where ${𝒟}_z$ is Heun’s operator in the variable $z$ : … ►

§31.10(ii) Type II

… ►

31.10.14 $$\left((t-z){𝒟}_s+(z-s){𝒟}_t+(s-t){𝒟}_z\right)𝒦=0,$$

ⓘ

Defines:

$𝒦(z;s,t)$: kernel (locally)

Symbols:

$z$: complex variable and ${𝒟}_z$: Heun’s operator

Permalink:

http://dlmf.nist.gov/31.10.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §31.10(ii), §31.10 and Ch.31

…

17: Bibliography H

… ►

An Euler-Maclaurin-type formula involving conjugate Bernoulli polynomials and an application to $\zeta (2m+1)$ . Commun. Appl. Anal. 1 (1), pp. 15–32.

ⓘ

External Links:

ISSN 1083-2564, MathReview (Pavel Guerzhoy), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

BibTeX

►

A Boole-type Formula involving Conjugate Euler Polynomials. In Charlemagne and his Heritage. 1200 Years of Civilization and Science in Europe, Vol. 2 (Aachen, 1995), P.L. Butzer, H. Th. Jongen, and W. Oberschelp (Eds.), pp. 361–375.

ⓘ

External Links:

ISBN 2-503-50674-7, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

BibTeX

… ►

An elementary approach to the hypergeometric shift operators of Opdam. Invent. Math. 103 (2), pp. 341–350.

ⓘ

External Links:

ISSN 0020-9910, Document, MathReview (Eric M. Opdam), ZentralBlatt

Referenced by:

§18.37(iii).

Encodings:

BibTeX

… ►

Definitions of Stieltjes Integrals of the Riemann Type. Amer. Math. Monthly 45 (5), pp. 265–278.

ⓘ

External Links:

ISSN 0002-9890, Document, Link, MathReview Entry

Referenced by:

§1.16(ix).

Encodings:

BibTeX

… ►

On a Painlevé-type boundary-value problem. Quart. J. Mech. Appl. Math. 37 (4), pp. 525–538.

ⓘ

External Links:

ISSN 0033-5614, Document, MathReview, ZentralBlatt

Referenced by:

§32.11(i), §32.17.

Encodings:

BibTeX

…

18: 3.9 Acceleration of Convergence

… ►It should be borne in mind that a sequence (series) transformation can be effective for one type of sequence (series) but may not accelerate convergence for another type. … ►

3.9.2 $$S=\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k{2}^{-k-1}{\mathrm{\Delta }}^k{a}_0,$$

ⓘ

Symbols:

$\mathrm{\Delta }$: forward difference operator and $S$: a convergent series

Permalink:

http://dlmf.nist.gov/3.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §3.9(ii), §3.9 and Ch.3

… ►Here $\mathrm{\Delta }$ is the forward difference operator: ►

3.9.3 $${\mathrm{\Delta }}^k{a}_0={\mathrm{\Delta }}^{k-1}{a}_1-{\mathrm{\Delta }}^{k-1}{a}_0,$$ $k=1,2,\mathrm{\dots }$.

ⓘ

Symbols:

$\mathrm{\Delta }$: forward difference operator

A&S Ref:

3.6.27

Permalink:

http://dlmf.nist.gov/3.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §3.9(ii), §3.9 and Ch.3

… ►

3.9.4 $${\mathrm{\Delta }}^k{a}_0=\sum _{m=0}^k{(-1)}^m\left(\genfrac{}{}{0.0pt}{}{k}{m}\right)a_{k-m}.$$

ⓘ

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient and $\mathrm{\Delta }$: forward difference operator

A&S Ref:

3.6.27

Permalink:

http://dlmf.nist.gov/3.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §3.9(ii), §3.9 and Ch.3

…

19: Bibliography G

… ►

An inequality of Turán type for Jacobi polynomials. Proc. Amer. Math. Soc. 32, pp. 435–439.

ⓘ

External Links:

ISSN 0002-9939, Document, MathReview (A. E. Danese), ZentralBlatt

Referenced by:

§18.14(ii).

Encodings:

BibTeX

►

Formulas of the Dirichlet-Mehler Type. In Fractional Calculus and its Applications, B. Ross (Ed.), Lecture Notes in Math., Vol. 457, pp. 207–215.

ⓘ

External Links:

MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§18.10(i).

Encodings:

BibTeX

… ►

Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.

ⓘ

Notes:

For remark see same journal 24(1998), p. 355.

External Links:

Document, ISSN 0098-3500, GAMS (module 726; package TOMS), ZentralBlatt

Referenced by:

2nd item, §3.5(v), §3.5(v).

Encodings:

BibTeX

… ►

Generalized Functions. Vol. 1: Properties and Operations. Academic Press, New York.

ⓘ

Notes:

Translated from the Russian by Eugene Saletan. A paperbound reprinting by the same publisher appeared in 1977

External Links:

MathReview, ZentralBlatt

Referenced by:

§2.6(i).

Encodings:

BibTeX

… ►

Riccati equations and convolution formulae for functions of Rayleigh type. J. Phys. A 33 (7), pp. 1363–1368.

ⓘ

External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§10.21(xiii).

Encodings:

BibTeX

…

20: 2.9 Difference Equations

… ►

2.9.2 $${\mathrm{\Delta }}^{2}w(n)+(2+f(n))\mathrm{\Delta }w(n)+(1+f(n)+g(n))w(n)=0,$$ $n=0,1,2,\mathrm{\dots }$,

ⓘ

Symbols:

$\mathrm{\Delta }$: forward difference operator, $f(n)$: function, $g(n)$: function, $n$: nonnegative integer and $w(n)$: solution

Permalink:

http://dlmf.nist.gov/2.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §2.9(i), §2.9 and Ch.2

►in which $\mathrm{\Delta }$ is the forward difference operator (§3.6(i)). … ►

§2.9(iii) Other Approximations

…