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1: 30.13 Wave Equation in Prolate Spheroidal Coordinates

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§30.13(ii) Metric Coefficients

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30.13.3 $$h_{\xi }^2={\left(\frac{\partial x}{\partial \xi }\right)}^2+{\left(\frac{\partial y}{\partial \xi }\right)}^2+{\left(\frac{\partial z}{\partial \xi }\right)}^2=\frac{c^2({\xi }^2-{\eta }^2)}{{\xi }^2-1},$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: complex variable, $x$: real variable, $y$: real variable, $\xi$: prolate spheroidal coordinate, $\eta$: prolate spheroidal coordinate, $c$: positive constant and $h_{\xi },h_{\eta },h_{\varphi }$: metric coefficients

A&S Ref:

21.4.2

Permalink:

http://dlmf.nist.gov/30.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §30.13(ii), §30.13 and Ch.30

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30.13.4 $$h_{\eta }^2={\left(\frac{\partial x}{\partial \eta }\right)}^2+{\left(\frac{\partial y}{\partial \eta }\right)}^2+{\left(\frac{\partial z}{\partial \eta }\right)}^2=\frac{c^2({\xi }^2-{\eta }^2)}{1-{\eta }^2},$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: complex variable, $x$: real variable, $y$: real variable, $\xi$: prolate spheroidal coordinate, $\eta$: prolate spheroidal coordinate, $c$: positive constant and $h_{\xi },h_{\eta },h_{\varphi }$: metric coefficients

Permalink:

http://dlmf.nist.gov/30.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §30.13(ii), §30.13 and Ch.30

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30.13.5 $$h_{\varphi }^2={\left(\frac{\partial x}{\partial \varphi }\right)}^2+{\left(\frac{\partial y}{\partial \varphi }\right)}^2+{\left(\frac{\partial z}{\partial \varphi }\right)}^2=c^2({\xi }^2-1)(1-{\eta }^2).$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: complex variable, $x$: real variable, $y$: real variable, $\xi$: prolate spheroidal coordinate, $\eta$: prolate spheroidal coordinate, $\varphi$: prolate spheroidal coordinate, $c$: positive constant and $h_{\xi },h_{\eta },h_{\varphi }$: metric coefficients

Permalink:

http://dlmf.nist.gov/30.13.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §30.13(ii), §30.13 and Ch.30

… ►In most applications the solution $w$ has to be a single-valued function of $(x,y,z)$, which requires $\mu =m$ (a nonnegative integer) and …

2: 30.14 Wave Equation in Oblate Spheroidal Coordinates

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§30.14(ii) Metric Coefficients

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30.14.3 $$h_{\xi }^2=\frac{c^2({\xi }^2+{\eta }^2)}{1+{\xi }^2},$$

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Symbols:

$h_{\xi },h_{\eta },h_{\varphi }$: metric coefficients, $\xi$: oblate spheroidal coordinate, $\eta$: oblate spheroidal coordinate and $c$: positive constant

A&S Ref:

21.4.3 (in different form)

Permalink:

http://dlmf.nist.gov/30.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §30.14(ii), §30.14 and Ch.30

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30.14.4 $$h_{\eta }^2=\frac{c^2({\xi }^2+{\eta }^2)}{1-{\eta }^2},$$

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Symbols:

$h_{\xi },h_{\eta },h_{\varphi }$: metric coefficients, $\xi$: oblate spheroidal coordinate, $\eta$: oblate spheroidal coordinate and $c$: positive constant

Permalink:

http://dlmf.nist.gov/30.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §30.14(ii), §30.14 and Ch.30

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30.14.5 $$h_{\varphi }^2=c^2({\xi }^2+1)(1-{\eta }^2).$$

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Symbols:

$h_{\xi },h_{\eta },h_{\varphi }$: metric coefficients, $\xi$: oblate spheroidal coordinate, $\eta$: oblate spheroidal coordinate, $\varphi$: oblate spheroidal coordinate and $c$: positive constant

Permalink:

http://dlmf.nist.gov/30.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §30.14(ii), §30.14 and Ch.30

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30.14.8 $$w_1(\xi )=a_1{S}_n^{m(1)}(\mathrm{i}\xi ,\gamma )+b_1{S}_n^{m(2)}(\mathrm{i}\xi ,\gamma ).$$

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Symbols:

$\mathrm{i}$: imaginary unit, $S_n^{m(j)}(z,\gamma )$: radial spheroidal wave function, $m$: nonnegative integer, $n\ge m$: integer degree, $b_j$: coefficients, $a_j$: coefficients, $\xi$: oblate spheroidal coordinate, $w_j$: solution to DE and ${\gamma }^2={\kappa }^2{c}^2$: parameter

Referenced by:

§30.14(v)

Permalink:

http://dlmf.nist.gov/30.14.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §30.14(iv), §30.14 and Ch.30

…

3: Bibliography K

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Pascal program for generating tables of $\mathrm{SU}(3)$ Clebsch-Gordan coefficients. Comput. Phys. Comm. 85 (1), pp. 82–88.

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External Links:

ISSN 0010-4655, Document, Code, MathReview Entry, ZentralBlatt

Referenced by:

6th item.

Encodings:

BibTeX

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Heat conduction and diffusion in a fluid medium with a periodic flow. I. Determination of the wave transfer coefficient in a tube, slot, and canal. Akad. Nauk SSSR. Žurnal Eksper. Teoret. Fiz. 21, pp. 964–978.

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External Links:

MathReview (N. A. Hall)

Referenced by:

§10.73(v).

Encodings:

BibTeX

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Gravitational field of a spinning mass as an example of algebraically special metrics. Phys. Rev. Lett. 11 (5), pp. 237–238.

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External Links:

Document, MathReview (C. Gilbert), ZentralBlatt

Referenced by:

§31.17(ii).

Encodings:

BibTeX

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Clebsch-Gordan coefficients for $\mathrm{SU}(2)$ and Hahn polynomials. Nieuw Arch. Wisk. (3) 29 (2), pp. 140–155.

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External Links:

ISSN 0028-9825, MathReview (Jiří Patera), ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

BibTeX

…

4: 26.3 Lattice Paths: Binomial Coefficients

§26.3 Lattice Paths: Binomial Coefficients

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§26.3(i) Definitions

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§26.3(ii) Generating Functions

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§26.3(iii) Recurrence Relations

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§26.3(iv) Identities

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5: 26.21 Tables

§26.21 Tables

►Abramowitz and Stegun (1964, Chapter 24) tabulates binomial coefficients $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$ for $m$ up to 50 and $n$ up to 25; extends Table 26.4.1 to $n=10$; tabulates Stirling numbers of the first and second kinds, $s(n,k)$ and $S(n,k)$, for $n$ up to 25 and $k$ up to $n$; tabulates partitions $p\left(n\right)$ and partitions into distinct parts $p(𝒟,n)$ for $n$ up to 500. … ►Goldberg et al. (1976) contains tables of binomial coefficients to $n=100$ and Stirling numbers to $n=40$.

6: 28.14 Fourier Series

… ►The coefficients satisfy ►

28.14.4 $$q{c}_{2m+2}-\left(a-{(\nu +2m)}^2\right)c_{2m}+q{c}_{2m-2}=0,$$ $a={\lambda }_{\nu }\left(q\right),c_{2m}=c_{2m}^{\nu }(q)$,

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Symbols:

${\lambda }_{\nu +2n}\left(q\right)$: eigenvalues of Mathieu equation, $m$: integer, $q=h^2$: parameter, $\nu$: complex parameter, $a$: parameter and $c_{2m}(q)$: coefficients

Referenced by:

item (d), item (d)

Permalink:

http://dlmf.nist.gov/28.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.14 and Ch.28

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28.14.5 $$\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{\left(c_{2m}^{\nu }(q)\right)}^2=1;$$

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Symbols:

$m$: integer, $q=h^2$: parameter, $\nu$: complex parameter and $c_{2m}(q)$: coefficients

Referenced by:

item (d)

Permalink:

http://dlmf.nist.gov/28.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.14 and Ch.28

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28.14.7 $$c_{-2m}^{-\nu }(q)=c_{2m}^{\nu }(q),$$

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Symbols:

$m$: integer, $q=h^2$: parameter, $\nu$: complex parameter and $c_{2m}(q)$: coefficients

Permalink:

http://dlmf.nist.gov/28.14.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §28.14 and Ch.28

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28.14.8 $$c_{2m}^{\nu }(-q)={(-1)}^m{c}_{2m}^{\nu }(q).$$

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Symbols:

$m$: integer, $q=h^2$: parameter, $\nu$: complex parameter and $c_{2m}(q)$: coefficients

Permalink:

http://dlmf.nist.gov/28.14.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §28.14 and Ch.28

…

7: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

§26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

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§26.4(i) Definitions

… ► $M_1$ is the multinominal coefficient (26.4.2): … ►

§26.4(ii) Generating Function

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§26.4(iii) Recurrence Relation

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8: 29.20 Methods of Computation

… ►Subsequently, formulas typified by (29.6.4) can be applied to compute the coefficients of the Fourier expansions of the corresponding Lamé functions by backward recursion followed by application of formulas typified by (29.6.5) and (29.6.6) to achieve normalization; compare §3.6. … ►A third method is to approximate eigenvalues and Fourier coefficients of Lamé functions by eigenvalues and eigenvectors of finite matrices using the methods of §§3.2(vi) and 3.8(iv). …The approximations converge geometrically (§3.8(i)) to the eigenvalues and coefficients of Lamé functions as $n\to \mathrm{\infty }$. … ►

§29.20(ii) Lamé Polynomials

… ►The corresponding eigenvectors yield the coefficients in the finite Fourier series for Lamé polynomials. …

9: 15.7 Continued Fractions

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15.7.1 $$\frac{𝐅(a,b;c;z)}{𝐅(a,b+1;c+1;z)}=t_0-\frac{u_{1}z}{t_1-}\frac{u_{2}z}{t_2-}\frac{u_{3}z}{t_3-}\mathrm{\cdots },$$

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Symbols:

$𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $t_n$: coefficient and $u_n$: coefficient

Permalink:

http://dlmf.nist.gov/15.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §15.7 and Ch.15

►where … ►

15.7.3 $$\frac{𝐅(a,b;c;z)}{𝐅(a,b+1;c+1;z)}=v_0-\frac{w_1}{v_1-}\frac{w_2}{v_2-}\frac{w_3}{v_3-}\mathrm{\cdots },$$

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Symbols:

$𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $v_n$: coefficient and $w_n$: coefficient

Permalink:

http://dlmf.nist.gov/15.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §15.7 and Ch.15

►where …

10: Joris Van der Jeugt

… ►His publications on Clebsch-Gordan coefficients, Racah coefficients, 3$nj$-coefficients and their relation to hypergeometric series are considered as standard and a review is part of the volume on Multivariable Special Functions in the ongoing Askey–Bateman book project. …