Gegenbauer polynomials

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21: 18.28 Askey–Wilson Class

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18.28.31 $$\underset{q\to 1}{lim}{C}_n(x;q^{\lambda }|q)=C_n^{(\lambda )}\left(x\right).$$

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Symbols:

$C_n(x;\beta |q)$: continuous $q$-ultraspherical polynomial, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $q$: real variable, $n$: nonnegative integer and $x$: real variable

Source:

Koekoek et al. (2010, (14.10.35))

Referenced by:

§18.28(x)

Permalink:

http://dlmf.nist.gov/18.28.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(x), §18.28(x), §18.28 and Ch.18

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22: Errata

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18.7.25

We included the case $n=0$.

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The Gegenbauer function $C_{\alpha }^{(\lambda )}\left(z\right)$, was labeled inadvertently as the ultraspherical (Gegenbauer) polynomial $C_n^{(\lambda )}\left(z\right)$. In order to resolve this inconsistency, this function now links correctly to its definition. This change affects Gegenbauer functions which appear in §§14.3(iv), 15.9(iii).

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The coefficient $A_n$ for $C_n^{(\lambda )}\left(x\right)$ in the first row of this table originally omitted the parentheses and was given as $\frac{2n+\lambda }{n+1}$, instead of $\frac{2(n+\lambda )}{n+1}$.

$p_n(x)$	$A_n$	$B_n$	$C_n$
$C_n^{(\lambda )}\left(x\right)$	$\frac{2(n+\lambda )}{n+1}$	$0$	$\frac{n+2\lambda -1}{n+1}$
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Reported 2010-09-16 by Kendall Atkinson.

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23: Bibliography C

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New proof of the addition theorem for Gegenbauer polynomials. SIAM J. Math. Anal. 2, pp. 347–351.

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External Links:

ISSN 1095-7154, Document, MathReview (Mary L. Boas), ZentralBlatt

Referenced by:

§18.18(ii).

Encodings:

BibTeX

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On a generalization of the generating function for Gegenbauer polynomials. Integral Transforms Spec. Funct. 24 (10), pp. 807–816.

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External Links:

ISSN 1065-2469, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§14.28(ii), §14.28(ii).

Encodings:

BibTeX

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24: 14.28 Sums

… ►For generalizations in terms of Gegenbauer and Jacobi polynomials, see Theorem 2. …

25: Bibliography K

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Orthogonal Polynomials on $n$-spheres: Gegenbauer, Jacobi and Heun. In Topics in Polynomials of One and Several Variables and their Applications, pp. 299–322.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§31.16(ii).

Encodings:

BibTeX

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26: Bibliography D

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On the real roots of Euler polynomials. Monatsh. Math. 106 (2), pp. 115–138.

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External Links:

ISSN 0026-9255, MathReview (H. M. Srivastava), ZentralBlatt

Referenced by:

§24.12(ii).

Encodings:

BibTeX

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On multiple zeros of Bernoulli polynomials. Acta Arith. 134 (2), pp. 149–155.

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External Links:

ISSN 0065-1036, Document, MathReview (Arnold M. Adelberg), ZentralBlatt

Referenced by:

§24.12(iv).

Encodings:

BibTeX

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Orthogonal polynomials and the construction of piecewise polynomial smooth wavelets. SIAM J. Math. Anal. 30 (5), pp. 1029–1056.

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External Links:

ISSN 0036-1410, Document, MathReview (Kathi Karima Selig), ZentralBlatt

Referenced by:

§16.4(iii).

Encodings:

BibTeX

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Uniform asymptotic expansions for Charlier polynomials. J. Approx. Theory 112 (1), pp. 93–133.

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External Links:

ISSN 0021-9045, Document, MathReview (Eberhard Wagner), ZentralBlatt

Referenced by:

§18.24.

Encodings:

BibTeX

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Nicholson-type Integrals for Products of Gegenbauer Functions and Related Topics. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), R. A. Askey (Ed.), pp. 353–374. Math. Res. Center, Univ. Wisconsin, Publ. No. 35.

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External Links:

MathReview (K. M. Saksena), ZentralBlatt

Referenced by:

(12.12.4), §12.12, §18.17(iii), §18.17(iii).

Encodings:

BibTeX

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27: 18.40 Methods of Computation

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§18.40(i) Computation of Polynomials

►Orthogonal polynomials can be computed from their explicit polynomial form by Horner’s scheme (§1.11(i)). … … ►The example chosen is inversion from the ${\alpha }_n,{\beta }_n$ for the weight function for the repulsive Coulomb–Pollaczek, RCP, polynomials of (18.39.50). … ►Further, exponential convergence in $N$, via the Derivative Rule, rather than the power-law convergence of the histogram methods, is found for the inversion of Gegenbauer, Attractive, as well as Repulsive, Coulomb–Pollaczek, and Hermite weights and zeros to approximate $w(x)$ for these OP systems on $x\in [-1,1]$ and $(-\mathrm{\infty },\mathrm{\infty })$ respectively, Reinhardt (2018), and Reinhardt (2021b), Reinhardt (2021a). …