Fourier transform of ultraspherical polynomials

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1: 18.17 Integrals

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Ultraspherical

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18.17.18 $${\int }_0^1{(1-x^2)}^{\lambda -\frac{1}{2}}{C}_{2n+1}^{(\lambda )}\left(x\right)\sin \left(xy\right)dx=\frac{{(-1)}^n\pi \mathrm{\Gamma }\left(2n+2\lambda +1\right)J_{2n+\lambda +1}\left(y\right)}{(2n+1)!\mathrm{\Gamma }\left(\lambda \right){(2y)}^{\lambda }}.$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $\sin z$: sine function, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $y$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.17(v)

Permalink:

http://dlmf.nist.gov/18.17.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

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2: 18.3 Definitions

§18.3 Definitions

… ►This table also includes the following special cases of Jacobi polynomials: ultraspherical, Chebyshev, and Legendre. … ►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). … ►It is also related to a discrete Fourier-cosine transform, see Britanak et al. (2007). …

3: Bibliography S

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Orthogonal polynomials arising in the numerical evaluation of inverse Laplace transforms. Math. Tables Aids Comput. 9 (52), pp. 164–177.

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External Links:

FullText, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§3.5(viii).

Encodings:

BibTeX

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Numerical evaluation of spherical Bessel transforms via fast Fourier transforms. J. Comput. Phys. 100 (2), pp. 294–296.

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External Links:

ISSN 0021-9991, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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A Guide to Distribution Theory and Fourier Transforms. Studies in Advanced Mathematics, CRC Press, Boca Raton, FL.

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External Links:

ISBN 0-8493-8273-4, MathReview (J. Horváth), ZentralBlatt

Referenced by:

§18.17(v).

Encodings:

BibTeX

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An Introduction to Basic Fourier Series. Developments in Mathematics, Vol. 9, Kluwer Academic Publishers, Dordrecht.

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External Links:

ISBN 1-4020-1221-7, MathReview (P. D. F. Ion), ZentralBlatt

Referenced by:

§17.3(i).

Encodings:

BibTeX

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On the relative extrema of ultraspherical polynomials. Boll. Un. Mat. Ital. (3) 5, pp. 125–127.

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§18.14(iii).

Encodings:

BibTeX

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4: 18.2 General Orthogonal Polynomials

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Kernel Polynomials

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§18.2(vi) Zeros

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§18.2(vii) Quadratic Transformations

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Sheffer Polynomials

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5: Bibliography L

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Optimal cylindrical and spherical Bessel transforms satisfying bound state boundary conditions. Comput. Phys. Comm. 99 (2-3), pp. 297–306.

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Notes:

Single-precision Fortran, maximum accuracy 8S.

External Links:

ISSN 0010-4655, Document, Code, ZentralBlatt

Referenced by:

4th item.

Encodings:

BibTeX

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An Introduction to Fourier Analysis and Generalised Functions. Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, New York.

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External Links:

ISBN 0521091284, MathReview (G. Temple), ZentralBlatt

Referenced by:

§1.16(ix), §1.16(v).

Encodings:

BibTeX

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Inequalities for ultraspherical polynomials and the gamma function. J. Approx. Theory 40 (2), pp. 115–120.

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External Links:

ISSN 0021-9045, Document, MathReview (B. R. Bhonsle), ZentralBlatt

Referenced by:

§18.14(i).

Encodings:

BibTeX

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Bessel transforms and rational extrapolation. Numer. Math. 47 (1), pp. 1–14.

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External Links:

ISSN 0029-599X, Document, MathReview (Vítĕzslav Veselý), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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Adjusted forms of the Fourier coefficient asymptotic expansion and applications in numerical quadrature. Math. Comp. 25 (113), pp. 87–104.

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External Links:

FullText, MathReview (J. Bryant), ZentralBlatt

Referenced by:

§2.3(iv).

Encodings:

BibTeX

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