Andrews–Askey sum

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11: 18.17 Integrals

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18.17.41_5 $${\int }_{-1}^1{C}_{\mathrm{\ell }}^{(\lambda )}\left(x\right)C_m^{(\lambda )}\left(x\right)C_n^{(\lambda )}\left(x\right){(1-x^2)}^{\lambda -\frac{1}{2}}dx=\frac{{\left(\lambda \right)}_{\frac{1}{2}\mathrm{\ell }+\frac{1}{2}m-\frac{1}{2}n}{\left(\lambda \right)}_{\frac{1}{2}m+\frac{1}{2}n-\frac{1}{2}\mathrm{\ell }}{\left(\lambda \right)}_{\frac{1}{2}n+\frac{1}{2}\mathrm{\ell }-\frac{1}{2}m}{\left(2\lambda \right)}_{\frac{1}{2}\mathrm{\ell }+\frac{1}{2}m+\frac{1}{2}n}\mathrm{\Gamma }\left(\lambda +\frac{1}{2}\right)\sqrt{\pi }}{(\frac{1}{2}\mathrm{\ell }+\frac{1}{2}m-\frac{1}{2}n)!(\frac{1}{2}m+\frac{1}{2}n-\frac{1}{2}\mathrm{\ell })!(\frac{1}{2}n+\frac{1}{2}\mathrm{\ell }-\frac{1}{2}m)!\mathrm{\Gamma }\left(\lambda +\frac{1}{2}\mathrm{\ell }+\frac{1}{2}m+\frac{1}{2}n+1\right)},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $\mathrm{\ell }$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Proved:

Andrews et al. (1999, Corollary 6.8.4); Straightforward from (18.18.22).(proved)

Referenced by:

§18.17(vii), Erratum (V1.2.0) Section 18.17

Permalink:

http://dlmf.nist.gov/18.17.E41_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

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

►provided that $\mathrm{\ell }+m+n$ is even and the sum of any two of $\mathrm{\ell },m,n$ is not less than the third; otherwise the integral is zero. … ►The case $x=1$ is a limit case of an integral for Jacobi polynomials; see Askey and Razban (1972). … ►provided that $\mathrm{\ell }+m+n$ is even and the sum of any two of $\mathrm{\ell },m,n$ is not less than the third; otherwise the integral is zero. …

12: Bibliography C

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Work Group of Computational Mathematics, University of Kassel, Germany.

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Notes:

Computer Algebra and Orthogonal Polynomials: a Web service using Maple for online generation of formulas and graphs of orthogonal polynomials belonging to the Askey scheme. For further information see Swarttouw (1997) and Koepf (1999).

External Links:

CAOP

Referenced by:

1st item.

Encodings:

BibTeX

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The hypergeometric function and the $R$-function near their branch points. Rend. Sem. Mat. Univ. Politec. Torino (Special Issue), pp. 63–89.

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Notes:

International conference on special functions: theory and computation (Turin, 1984)

External Links:

ISSN 0373-1243, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§19.27(vi).

Encodings:

BibTeX

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On the zeros of the Askey-Wilson polynomials, with applications to coding theory. SIAM J. Math. Anal. 18 (1), pp. 191–207.

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External Links:

ISSN 0036-1410, Document, MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

BibTeX

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Approximations and Complex Multiplication According to Ramanujan. In Ramanujan Revisited (Urbana-Champaign, Ill., 1987), G. E. Andrews, R. A. Askey, B. C. Bernd, K. G. Ramanathan, and R. A. Rankin (Eds.), pp. 375–472.

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External Links:

MathReview (Maurice Mignotte), ZentralBlatt

Referenced by:

§20.11(iii).

Encodings:

BibTeX

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Remarks on the full asymptotic expansion of Feynman parametrized integrals. Lett. Nuovo Cimento (2) 13 (8), pp. 310–312.

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External Links:

MathReview (R. A. Askey)

Referenced by:

§2.3(iii).

Encodings:

BibTeX

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