Gauss–Jacobi formula

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11: Bibliography I

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Two families of orthogonal polynomials related to Jacobi polynomials. Rocky Mountain J. Math. 21 (1), pp. 359–375.

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External Links:

ISSN 0035-7596, MathReview (Joaquin Bustoz), ZentralBlatt

Referenced by:

§18.30(i).

Encodings:

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Asymptotics of the Askey-Wilson and $q$-Jacobi polynomials. SIAM J. Math. Anal. 17 (6), pp. 1475–1482.

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External Links:

ISSN 0036-1410, Document, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§18.29.

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The method of isomonodromic deformations and relation formulas for the second Painlevé transcendent. Izv. Akad. Nauk SSSR Ser. Mat. 51 (4), pp. 878–892, 912 (Russian).

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Notes:

In Russian. English translation: Math. USSR-Izv. 31(1988), no. 1, pp. 193–207

External Links:

ISSN 0373-2436, Document, MathReview (Alexander A. Pankov), ZentralBlatt

Referenced by:

§32.11(iii).

Encodings:

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Connection formulae for the fourth Painlevé transcendent; Clarkson-McLeod solution. J. Phys. A 31 (17), pp. 4073–4113.

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External Links:

ISSN 0305-4470, Document, MathReview (Yoshitsugu Takei), ZentralBlatt

Referenced by:

§32.11(v).

Encodings:

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From Gauss to Painlevé: A Modern Theory of Special Functions. Aspects of Mathematics E, Vol. 16, Friedr. Vieweg & Sohn, Braunschweig, Germany.

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External Links:

ISBN 3-528-06355-6, MathReview (Takeshi Sasaki), ZentralBlatt

Referenced by:

§32.2(vi), §32.7(vii).

Encodings:

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12: 18.38 Mathematical Applications

… ►If the nodes in a quadrature formula with a positive weight function are chosen to be the zeros of the $n$th degree OP with the same weight function, and the interval of orthogonality is the same as the integration range, then the weights in the quadrature formula can be chosen in such a way that the formula is exact for all polynomials of degree not exceeding $2n-1$. … ►The Askey–Gasper inequality …For the generalized hypergeometric function ${}_3{}^{}F_2^{}$ see (16.2.1). … ►The $3j$ symbol (34.2.6), with an alternative expression as a terminating ${}_3{}^{}F_2^{}$ of unit argument, can be expressed in terms of Hahn polynomials (18.20.5) or, by (18.21.1), dual Hahn polynomials. … ►See Koornwinder (2007a, (3.13), (4.9), (4.10)) for explicit formulas. …

13: Bibliography K

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On the evaluation of the Gauss hypergeometric function. C. R. Acad. Bulgare Sci. 45 (6), pp. 35–36.

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External Links:

ISSN 0861-1459, MathReview, ZentralBlatt

Referenced by:

§15.15, §15.19(i).

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On formulas involving both the Bernoulli and Fibonacci numbers. Scripta Math. 23, pp. 27–35.

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External Links:

MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§24.15(iv).

Encodings:

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Jacobi polynomials. II. An analytic proof of the product formula. SIAM J. Math. Anal. 5, pp. 125–137.

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External Links:

Document, MathReview (R. P. Soni), ZentralBlatt

Referenced by:

§18.12, §18.17(ii), §18.18(vi).

Encodings:

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Jacobi polynomials. III. An analytic proof of the addition formula. SIAM. J. Math. Anal. 6, pp. 533–543.

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External Links:

Document, MathReview (G. Gasper), ZentralBlatt

Referenced by:

§18.18(ii).

Encodings:

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The addition formula for Laguerre polynomials. SIAM J. Math. Anal. 8 (3), pp. 535–540.

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External Links:

Document, MathReview (M. Mikolas), ZentralBlatt

Referenced by:

(18.14.8), §18.14(i), §18.17(ii), §18.18(ii).

Encodings:

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14: 18.17 Integrals

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Jacobi

… ►For formulas for Jacobi and Laguerre polynomials analogous to (18.17.5) and (18.17.6), see Koornwinder (1974, 1977). … ►For similar formulas for ultraspherical polynomials see Durand (1975), and for Jacobi and Laguerre polynomials see Durand (1978). … ►

Jacobi

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Jacobi

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15: 18.26 Wilson Class: Continued

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18.26.2 $$S_n(y^2;a,b,c)={\left(a+b\right)}_n{\left(a+c\right)}_n{}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{-n,a+iy,a-iy}{a+b,a+c};1).$$

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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), $S_n(x;a,b,c)$: continuous dual Hahn polynomial, $y$: real variable and $n$: nonnegative integer

Referenced by:

§18.26(i), §18.26(ii), §18.26(iv)

Permalink:

http://dlmf.nist.gov/18.26.E2

Encodings:

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See also:

Annotations for §18.26(i), §18.26 and Ch.18

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Wilson $\to$ Jacobi

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18.26.7 $$\underset{t\to \mathrm{\infty }}{lim}\frac{W_n(\frac{1}{2}(1-x)t^2;\frac{1}{2}\alpha +\frac{1}{2},\frac{1}{2}\alpha +\frac{1}{2},\frac{1}{2}\beta +\frac{1}{2}+it,\frac{1}{2}\beta +\frac{1}{2}-it)}{t^{2n}n!}=P_n^{(\alpha ,\beta )}\left(x\right).$$

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Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $W_n(x;a,b,c,d)$: Wilson polynomial, $!$: factorial (as in $n!$), $t$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.26(ii)

Permalink:

http://dlmf.nist.gov/18.26.E7

Encodings:

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See also:

Annotations for §18.26(ii), §18.26(ii), §18.26 and Ch.18

… ►See Koekoek et al. (2010, Chapter 9) for further formulas. … ►For the hypergeometric function ${}_2{}^{}F_1^{}$ see §§15.1 and 15.2(i). …

16: 18.2 General Orthogonal Polynomials

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§18.2(v) Christoffel–Darboux Formula

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Confluent Form

… ►For usage of the zeros of an OP in Gauss quadrature see §3.5(v). … ►

Degree lowering and raising differentiation formulas and structure relations

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17: Bibliography C

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A quadrature formula for the Hankel transform. Numer. Algorithms 9 (2), pp. 343–354.

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External Links:

ISSN 1017-1398, Document, MathReview (V. I. Polovinkin), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

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Gauss hypergeometric representations of the Ferrers function of the second kind. SIGMA Symmetry Integrability Geom. Methods Appl. 17, pp. Paper 053, 33.

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External Links:

Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

(14.13.2), §14.3(iii), §14.3(iii).

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An encyclopaedia of cubature formulas. J. Complexity 19 (3), pp. 445–453.

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Notes:

Numerical integration and its complexity (Oberwolfach, 2001)

External Links:

ISSN 0885-064X, Document, Web Page, MathReview, ZentralBlatt

Referenced by:

§3.5(x).

Encodings:

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The arithmetic-geometric mean of Gauss. Enseign. Math. (2) 30 (3-4), pp. 275–330.

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External Links:

ISSN 0013-8584, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§19.8(i).

Encodings:

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Gauss and the arithmetic-geometric mean. Notices Amer. Math. Soc. 32 (2), pp. 147–151.

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External Links:

ISSN 0002-9920, MathReview (Willard Parker)

Referenced by:

§19.8(i).

Encodings:

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