associated Jacobi polynomials

SearchAdvancedHelp

(0.005 seconds)

1—10 of 23 matching pages

1: 18.30 Associated OP’s

… ►

§18.30(i) Associated Jacobi Polynomials

… ►

18.30.4 $$P_n^{(\alpha ,\beta )}(x;c)=p_n(x;c),$$ $n=0,1,\mathrm{\dots }$,

ⓘ

Defines:

$P_n^{(\alpha ,\beta )}(x;c)$: associated Jacobi polynomial

Symbols:

$p_n(x)$: polynomial of degree $n$ , $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.30

Permalink:

http://dlmf.nist.gov/18.30.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.30(i), §18.30 and Ch.18

… ►

18.30.5 $$\frac{{(-1)}^n{\left(\alpha +\beta +c+1\right)}_{n}n!P_n^{(\alpha ,\beta )}(x;c)}{{\left(\alpha +\beta +2c+1\right)}_n{\left(\beta +c+1\right)}_n}=\sum _{\mathrm{\ell }=0}^n\frac{{\left(-n\right)}_{\mathrm{\ell }}{\left(n+\alpha +\beta +2c+1\right)}_{\mathrm{\ell }}}{{\left(c+1\right)}_{\mathrm{\ell }}{\left(\beta +c+1\right)}_{\mathrm{\ell }}}{\left(\frac{1}{2}x+\frac{1}{2}\right)}^{\mathrm{\ell }}{}_4{}^{}F_3^{}(\genfrac{}{}{0pt}{}{\mathrm{\ell }-n,n+\mathrm{\ell }+\alpha +\beta +2c+1,\beta +c,c}{\beta +\mathrm{\ell }+c+1,\mathrm{\ell }+c+1,\alpha +\beta +2c};1),$$

ⓘ

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $P_n^{(\alpha ,\beta )}(x;c)$: associated Jacobi polynomial, $!$: factorial (as in $n!$), $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.30

Permalink:

http://dlmf.nist.gov/18.30.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §18.30(i), §18.30 and Ch.18

… ►For corresponding corecursive associated Jacobi polynomials, corecursive associated polynomials being discussed in §18.30(vii), see Letessier (1995). … ►

18.30.6 $$P_n(x;c)=P_n^{(0,0)}(x;c),$$ $n=0,1,\mathrm{\dots }$.

ⓘ

Defines:

$P_n(x;c)$: associated Legendre polynomial

Symbols:

$P_n^{(\alpha ,\beta )}(x;c)$: associated Jacobi polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.30

Permalink:

http://dlmf.nist.gov/18.30.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.30(ii), §18.30 and Ch.18

…

2: Bibliography W

… ►

Explicit formulas for the associated Jacobi polynomials and some applications. Canad. J. Math. 39 (4), pp. 983–1000.

ⓘ

External Links:

ISSN 0008-414X, MathReview (W. A. Al-Salam), ZentralBlatt

Referenced by:

§18.30(i), §18.30.

Encodings:

BibTeX

…

3: 18.37 Classical OP’s in Two or More Variables

… ►In several variables they occur, for $q=1$, as Jack polynomials and also as Jacobi polynomials associated with root systems; see Macdonald (1995, Chapter VI, §10), Stanley (1989), Kuznetsov and Sahi (2006, Part 1), Heckman (1991). …

4: Bibliography L

… ►

Co-recursive associated Jacobi polynomials. J. Comput. Appl. Math. 57 (1-2), pp. 203–213.

ⓘ

External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§18.30(i).

Encodings:

BibTeX

…

5: Bibliography E

… ►

Uniform asymptotic expansions of the Jacobi polynomials and an associated function. Math. Comp. 25 (114), pp. 309–315.

ⓘ

External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§18.15(i).

Encodings:

BibTeX

…

6: 18.38 Mathematical Applications

… ►Dunkl type operators and nonsymmetric polynomials have been associated with various other families in the Askey scheme and $q$-Askey scheme, in particular with Wilson polynomials, see Groenevelt (2007), and with Jacobi polynomials, see Koornwinder and Bouzeffour (2011, §7). …

7: 18.3 Definitions

§18.3 Definitions

►The classical OP’s comprise the Jacobi, Laguerre and Hermite polynomials. … ►For expressions of ultraspherical, Chebyshev, and Legendre polynomials in terms of Jacobi polynomials, see §18.7(i). …For explicit power series coefficients up to $n=12$ for these polynomials and for coefficients up to $n=6$ for Jacobi and ultraspherical polynomials see Abramowitz and Stegun (1964, pp. 793–801). … ►Legendre polynomials are special cases of Legendre functions, Ferrers functions, and associated Legendre functions (§14.7(i)). …

8: Bibliography R

… ►

A non-negative representation of the linearization coefficients of the product of Jacobi polynomials. Canad. J. Math. 33 (4), pp. 915–928.

ⓘ

External Links:

ISSN 0008-414X, Document, Link, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.18(v).

Encodings:

BibTeX

►

The Associated Classical Orthogonal Polynomials. In Special Functions 2000: Current Perspective and Future Directions (Tempe, AZ), NATO Sci. Ser. II Math. Phys. Chem., Vol. 30, pp. 255–279.

ⓘ

External Links:

MathReview (Wadim Zudilin), ZentralBlatt

Referenced by:

§18.30(viii).

Encodings:

BibTeX

… ►

Erratum to:Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging. Computing in Science and Engineering 23 (4), pp. 91.

ⓘ

External Links:

ISSN 1521-9615, Document, Link

Referenced by:

§18.39(iv), §18.40(ii).

Encodings:

BibTeX

►

Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging. Computing in Science and Engineering 23 (3), pp. 56–64.

ⓘ

External Links:

ISSN 1521-9615, Document, Link

Referenced by:

§18.39(iv), §18.40(ii).

Encodings:

BibTeX

… ►

Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications. Contemporary Mathematics, Vol. 138, American Mathematical Society, Providence, RI.

ⓘ

External Links:

ISBN 0-8218-5159-4, MathReview, ZentralBlatt

Referenced by:

Ch.35, Profile Donald St. P. Richards.

Encodings:

BibTeX

…

9: Errata

… ►In 2016, on the advice of the senior associate editors, is was decided to expand Chapter 18 (Orthogonal Polynomials (OP)). This release is the result of that decision and it includes, among other new material, enlarged sections on associated OP’s, Pollaczek polynomials and physical applications. … ►We have also incorporated material on continuous $q$-Jacobi polynomials, and several new limit transitions. We have significantly expanded the section on associated orthogonal polynomials, including expanded properties of associated Laguerre, Hermite, Meixner–Pollaczek, and corecursive orthogonal and numerator and denominator orthogonal polynomials. … ►

Special cases of normalization of Jacobi polynomials for which the general formula is undefined have been stated explicitly in Table 18.3.1.

…

10: 14.31 Other Applications

… ►

§14.31(ii) Conical Functions

… ►The conical functions and Mehler–Fock transform generalize to Jacobi functions and the Jacobi transform; see Koornwinder (1984a) and references therein. ►

§14.31(iii) Miscellaneous

►Many additional physical applications of Legendre polynomials and associated Legendre functions include solution of the Helmholtz equation, as well as the Laplace equation, in spherical coordinates (Temme (1996b)), quantum mechanics (Edmonds (1974)), and high-frequency scattering by a sphere (Nussenzveig (1965)). … ►