zonal spherical harmonics

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21: 18.39 Applications in the Physical Sciences

… ►argument a) The Harmonic Oscillator … ►By (1.5.17) the first term in (18.39.21), which is the quantum kinetic energy operator $T_e$, can be written in spherical coordinates $r,\theta ,\varphi$ as …The eigenfunctions of ${\mathrm{L}}^2$ are the spherical harmonics $Y_{l,m_l}(\theta ,\varphi )$ with eigenvalues ${\mathrm{\hslash }}^{2}l(l+1)$, each with degeneracy $2l+1$ as $m_l=-l,-l+1,\mathrm{\dots },l$. … … ►

c) Spherical Radial Coulomb Wave Functions

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22: 10.47 Definitions and Basic Properties

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Equation (10.47.1)

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Equation (10.47.2)

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§10.47(iv) Interrelations

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23: 10.49 Explicit Formulas

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§10.49(i) Unmodified Functions

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§10.49(ii) Modified Functions

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§10.49(iii) Rayleigh’s Formulas

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§10.49(iv) Sums or Differences of Squares

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${\left({𝗂}_0^{(1)}\left(z\right)\right)}^2-{\left({𝗂}_0^{(2)}\left(z\right)\right)}^2=-z^{-2},$

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24: 35.1 Special Notation

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$a,b$	complex variables.
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$Z_{\kappa }\left(𝐓\right)$	zonal polynomials.

… ►Related notations for the Bessel functions are ${𝒥}_{\nu +\frac{1}{2}(m+1)}(𝐓)=A_{\nu }\left(𝐓\right)/A_{\nu }\left(𝟎\right)$ (Faraut and Korányi (1994, pp. 320–329)), $K_m(0,\mathrm{\dots },0,\nu |𝐒,𝐓)={\left|𝐓\right|}^{\nu }{B}_{\nu }\left(𝐒𝐓\right)$ (Terras (1988, pp. 49–64)), and ${𝒦}_{\nu }(𝐓)={\left|𝐓\right|}^{\nu }{B}_{\nu }\left(𝐒𝐓\right)$ (Faraut and Korányi (1994, pp. 357–358)).

25: 14.18 Sums

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Zonal Harmonic Series

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26: 17.17 Physical Applications

… ►See Kassel (1995). … ►It involves $q$-generalizations of exponentials and Laguerre polynomials, and has been applied to the problems of the harmonic oscillator and Coulomb potentials. …

27: 10.53 Power Series

§10.53 Power Series

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10.53.3 $${𝗂}_n^{(1)}\left(z\right)=z^n\sum _{k=0}^{\mathrm{\infty }}\frac{{(\frac{1}{2}{z}^2)}^k}{k!(2n+2k+1)!!},$$

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Symbols:

$!!$: double factorial, $!$: factorial (as in $n!$), ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $n$: integer, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

10.2.5

Referenced by:

§10.53

Permalink:

http://dlmf.nist.gov/10.53.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.53 and Ch.10

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10.53.4 $${𝗂}_n^{(2)}\left(z\right)=\frac{{(-1)}^n}{z^{n+1}}\sum _{k=0}^n\frac{(2n-2k-1)!!{(-\frac{1}{2}{z}^2)}^k}{k!}+\frac{1}{z^{n+1}}\sum _{k=n+1}^{\mathrm{\infty }}\frac{{(\frac{1}{2}{z}^2)}^k}{k!(2k-2n-1)!!}.$$

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Symbols:

$!!$: double factorial, $!$: factorial (as in $n!$), ${𝗂}_n^{(2)}\left(z\right)$: modified spherical Bessel function, $n$: integer, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

10.2.6

Referenced by:

§10.53

Permalink:

http://dlmf.nist.gov/10.53.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.53 and Ch.10

►For ${𝗁}_n^{(1)}\left(z\right)$ and ${𝗁}_n^{(2)}\left(z\right)$ combine (10.47.10), (10.53.1), and (10.53.2). For ${𝗄}_n\left(z\right)$ combine (10.47.11), (10.53.3), and (10.53.4).

28: 10.56 Generating Functions

§10.56 Generating Functions

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10.56.1 $$\frac{\cos \sqrt{z^2-2zt}}{z}=\frac{\cos z}{z}+\sum _{n=1}^{\mathrm{\infty }}\frac{t^n}{n!}{𝗃}_{n-1}\left(z\right),$$

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Symbols:

$\cos z$: cosine function, $!$: factorial (as in $n!$), ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable

Referenced by:

§10.56, Ch.10

Permalink:

http://dlmf.nist.gov/10.56.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.56 and Ch.10

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10.56.2 $$\frac{\sin \sqrt{z^2-2zt}}{z}=\frac{\sin z}{z}+\sum _{n=1}^{\mathrm{\infty }}\frac{t^n}{n!}{𝗒}_{n-1}\left(z\right).$$

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Symbols:

$!$: factorial (as in $n!$), $\sin z$: sine function, ${𝗒}_n\left(z\right)$: spherical Bessel function of the second kind, $n$: integer and $z$: complex variable

Referenced by:

§10.56

Permalink:

http://dlmf.nist.gov/10.56.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.56 and Ch.10

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10.56.3 $$\frac{\cosh \sqrt{z^2+2\mathrm{i}zt}}{z}=\frac{\cosh z}{z}+\sum _{n=1}^{\mathrm{\infty }}\frac{{(\mathrm{i}t)}^n}{n!}{𝗂}_{n-1}^{(1)}\left(z\right),$$

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Symbols:

$!$: factorial (as in $n!$), $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $n$: integer and $z$: complex variable

Referenced by:

§10.56

Permalink:

http://dlmf.nist.gov/10.56.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.56 and Ch.10

►

10.56.4 $$\frac{\sinh \sqrt{z^2+2\mathrm{i}zt}}{z}=\frac{\sinh z}{z}+\sum _{n=1}^{\mathrm{\infty }}\frac{{(\mathrm{i}t)}^n}{n!}{𝗂}_{n-1}^{(2)}\left(z\right),$$

ⓘ

Symbols:

$!$: factorial (as in $n!$), $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, ${𝗂}_n^{(2)}\left(z\right)$: modified spherical Bessel function, $n$: integer and $z$: complex variable

Referenced by:

§10.56

Permalink:

http://dlmf.nist.gov/10.56.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.56 and Ch.10

…

29: 10.60 Sums

§10.60 Sums

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§10.60(i) Addition Theorems

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§10.60(ii) Duplication Formulas

… ►For further sums of series of spherical Bessel functions, or modified spherical Bessel functions, see §6.10(ii), Luke (1969b, pp. 55–58), Vavreck and Thompson (1984), Harris (2000), and Rottbrand (2000). ►

§10.60(iv) Compendia

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30: 10.51 Recurrence Relations and Derivatives

… ►Let $f_n(z)$ denote any of ${𝗃}_n\left(z\right)$, ${𝗒}_n\left(z\right)$, ${𝗁}_n^{(1)}\left(z\right)$, or ${𝗁}_n^{(2)}\left(z\right)$. … ►

$$n{f}_{n-1}(z)-(n+1)f_{n+1}(z)=(2n+1)f_n^{\prime }(z),$$ $n=1,2,\mathrm{\dots }$,

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$$f_n^{\prime }(z)=-f_{n+1}(z)+(n/z)f_n(z),$$ $n=0,1,\mathrm{\dots }.$

… ►Let $g_n(z)$ denote ${𝗂}_n^{(1)}\left(z\right)$, ${𝗂}_n^{(2)}\left(z\right)$, or ${(-1)}^n$ ${𝗄}_n\left(z\right)$. Then …