Gegenbauer%20addition%20theorem

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2: 10.23 Sums

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§10.23(ii) Addition Theorems

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Neumann’s Addition Theorem

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Graf’s and Gegenbauer’s Addition Theorems

… ►where

C^{(\nu)}_{k}\left(\cos\alpha\right)

is Gegenbauer’s polynomial (§18.3). … ► …

3: Bibliography K

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E. G. Kalnins and W. Miller (1993) 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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R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

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External Links:: ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

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M. K. Kerimov (1980) Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

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External Links:: Document, MathReview (Matti Jutila), ZentralBlatt
Referenced by:: §25.18(i).
Encodings:: BibTeX

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A. V. Kitaev and A. H. Vartanian (2004) Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

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External Links:: ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt
Referenced by:: §32.11(iv).
Encodings:: BibTeX

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T. H. Koornwinder (2009) The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.

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External Links:: ISSN 1382-4090, Document, FullText, MathReview (José Luis López), ZentralBlatt
Referenced by:: §18.26(v), §18.26(v).
Encodings:: BibTeX

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►Addition theorems provide important connections between Mathieu functions with different parameters and in different coordinate systems. They are analogous to the addition theorems for Bessel functions (§10.23(ii)) and modified Bessel functions (§10.44(ii)). …

7: 18.1 Notation

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Ultraspherical (or Gegenbauer): $C^{(\lambda)}_{n}\left(x\right)$ .

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Associated OP’s are denoted via addition of the letter $c$ at the end of the listing of parameters in their usual notations.

… ►In Szegő (1975, §4.7) the ultraspherical polynomials

C^{(\lambda)}_{n}\left(x\right)

are denoted by

P_{n}^{(\lambda)}(x)

. The ultraspherical polynomials will not be considered for

\lambda=0

. They are defined in the literature by

C^{(0)}_{0}\left(x\right)=1

and …

8: Bibliography C

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B. C. Carlson (1971) 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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R. Chelluri, L. B. Richmond, and N. M. Temme (2000) Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.

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External Links:: ISSN 0174-4747, MathReview (F. T. Howard), ZentralBlatt
Referenced by:: §26.8(vii).
Encodings:: BibTeX

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H. S. Cohl (2013a) Fourier, Gegenbauer and Jacobi expansions for a power-law fundamental solution of the polyharmonic equation and polyspherical addition theorems. SIGMA Symmetry Integrability Geom. Methods Appl. 9, pp. Paper 042, 26.

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External Links:: ISSN 1815-0659, Document, MathReview (Heinrich Begehr), ZentralBlatt
Referenced by:: §14.28(ii), §14.28(ii).
Encodings:: BibTeX

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H. S. Cohl (2013b) 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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M. Colman, A. Cuyt, and J. Van Deun (2011) Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

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External Links:: ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §13.29(v), §15.19(v).
Encodings:: BibTeX

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9: Bibliography D

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C. de la Vallée Poussin (1896b) Recherches analytiques sur la théorie des nombres premiers. Deuxième partie. Les fonctions de Dirichlet et les nombres premiers de la forme linéaire

Mx+N

. Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).

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Notes:: Reprinted in Collected works/Oeuvres scientifiques, Vol. I, pp. 309–425, Académie Royale de Belgique, Brussels, 2000.
External Links:: MathReview, ZentralBlatt
Referenced by:: §25.10(i), §27.11, §27.2(i).
Encodings:: BibTeX

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S. C. Dhar (1940) Note on the addition theorem of parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 4, pp. 29–30.

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External Links:: MathReview (M. C. Gray), ZentralBlatt
Referenced by:: §12.13(ii).
Encodings:: BibTeX

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B. Döring (1966) Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

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Notes:: For a numerical error in one of the zeros see Amos (1985).
External Links:: ISSN 0025-5718, FullText, MathReview, ZentralBlatt
Referenced by:: 2nd item.
Encodings:: BibTeX

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T. M. Dunster (1989) Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.

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Notes:: (This reference has several typographical errors.)
External Links:: ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §13.21(ii), §13.21(iii), §13.21(iv).
Encodings:: BibTeX

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L. Durand (1975) 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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10: 18.10 Integral Representations

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18.10.1

\frac{P^{(\alpha,\alpha)}_{n}\left(\cos\theta\right)}{P^{(\alpha,\alpha)}_{n}% \left(1\right)}=\frac{C^{(\alpha+\frac{1}{2})}_{n}\left(\cos\theta\right)}{C^{% (\alpha+\frac{1}{2})}_{n}\left(1\right)}=\frac{2^{\alpha+\frac{1}{2}}\Gamma% \left(\alpha+1\right)}{{\pi}^{\frac{1}{2}}\Gamma\left(\alpha+\frac{1}{2}\right% )}(\sin\theta)^{-2\alpha}\int_{0}^{\theta}\frac{\cos\left((n+\alpha+\tfrac{1}{% 2})\phi\right)}{(\cos\phi-\cos\theta)^{-\alpha+\frac{1}{2}}}\,\mathrm{d}\phi,

0<\theta<\pi

\alpha>-\tfrac{1}{2}

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial and $n$ : nonnegative integer
A&S Ref:: 22.10.11
Referenced by:: §18.10(i), §18.10(i)
Permalink:: http://dlmf.nist.gov/18.10.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §18.10(i), §18.10(i), §18.10 and Ch.18

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18.10.4

{\frac{P^{(\alpha,\alpha)}_{n}\left(\cos\theta\right)}{P^{(\alpha,\alpha)}_{n}% \left(1\right)}=\frac{C^{(\alpha+\frac{1}{2})}_{n}\left(\cos\theta\right)}{C^{% (\alpha+\frac{1}{2})}_{n}\left(1\right)}}=\frac{\Gamma\left(\alpha+1\right)}{{% \pi}^{\frac{1}{2}}\Gamma{(\alpha+\tfrac{1}{2})}}\*{\int_{0}^{\pi}(\cos\theta+i% \sin\theta\cos\phi)^{n}\*(\sin\phi)^{2\alpha}\,\mathrm{d}\phi},

\alpha>-\frac{1}{2}

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial and $n$ : nonnegative integer
A&S Ref:: 22.10.10
Referenced by:: §18.10(ii)
Permalink:: http://dlmf.nist.gov/18.10.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §18.10(ii), §18.10(ii), §18.10 and Ch.18

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Table 18.10.1: Classical OP’s: contour integral representations (18.10.8).

► ►►►

$p_{n}(x)$	$g_{0}(x)$	$g_{1}(z,x)$	$g_{2}(z,x)$	$c$	Conditions
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$C^{(\lambda)}_{n}\left(x\right)$	$1$	$z^{-1}$	$(1-2xz+z^{2})^{-\lambda}$	$0$	${\mathrm{e}}^{\pm\mathrm{i}\theta}$ outside $C$ (where $x=\cos\theta$ ).
…

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Gegenbauer%20addition%20theorem

1—10 of 259 matching pages

1: 10.44 Sums

§10.44(i) Multiplication Theorem

§10.44(ii) Addition Theorems

Neumann’s Addition Theorem

Graf’s and Gegenbauer’s Addition Theorems

2: 10.23 Sums

§10.23(ii) Addition Theorems

Neumann’s Addition Theorem

Graf’s and Gegenbauer’s Addition Theorems

3: Bibliography K

4: 18.18 Sums

§18.18(ii) Addition Theorems

Ultraspherical

Legendre

Laguerre

Hermite

5: 20 Theta Functions

Chapter 20 Theta Functions

6: 28.27 Addition Theorems

§28.27 Addition Theorems

7: 18.1 Notation

8: Bibliography C

9: Bibliography D

10: 18.10 Integral Representations