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11: 28.23 Expansions in Series of Bessel Functions

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28.23.6 $${\mathrm{Mc}}_{2m}^{(j)}(z,h)={(-1)}^m{\left({\mathrm{ce}}_{2m}(0,h^2)\right)}^{-1}\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{\ell }}{A}_{2\mathrm{\ell }}^{2m}(h^2){𝒞}_{2\mathrm{\ell }}^{(j)}(2h\cosh z),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cosh z$: hyperbolic cosine function, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $A_m(q)$: Fourier coefficient

A&S Ref:

20.6.12

Referenced by:

§28.23

Permalink:

http://dlmf.nist.gov/28.23.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

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28.23.8 $${\mathrm{Mc}}_{2m+1}^{(j)}(z,h)={(-1)}^m{\left({\mathrm{ce}}_{2m+1}(0,h^2)\right)}^{-1}\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{\ell }}{A}_{2\mathrm{\ell }+1}^{2m+1}(h^2){𝒞}_{2\mathrm{\ell }+1}^{(j)}(2h\cosh z),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cosh z$: hyperbolic cosine function, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.23.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

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28.23.10 $${\mathrm{Ms}}_{2m+1}^{(j)}(z,h)={(-1)}^m{\left({\mathrm{se}}_{2m+1}^{\prime }(0,h^2)\right)}^{-1}\tanh z\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{\ell }}(2\mathrm{\ell }+1)B_{2\mathrm{\ell }+1}^{2m+1}(h^2){𝒞}_{2\mathrm{\ell }+1}^{(j)}(2h\cosh z),$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.23.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

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28.23.12 $${\mathrm{Ms}}_{2m+2}^{(j)}(z,h)={(-1)}^m{\left({\mathrm{se}}_{2m+2}^{\prime }(0,h^2)\right)}^{-1}\tanh z\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{\ell }}(2\mathrm{\ell }+2)B_{2\mathrm{\ell }+2}^{2m+2}(h^2){𝒞}_{2\mathrm{\ell }+2}^{(j)}(2h\cosh z),$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.23.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

… ►When $j=2,3,4$ the series in the even-numbered equations converge for $\mathrm{\Re }z>0$ and $|\cosh z|>1$, and the series in the odd-numbered equations converge for $\mathrm{\Re }z>0$ and $|\sinh z|>1$. …

12: 33.5 Limiting Forms for Small $\rho$, Small $|\eta |$, or Large $\mathrm{\ell }$

§33.5 Limiting Forms for Small $\rho$, Small $|\eta |$, or Large $\mathrm{\ell }$

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$F_{\mathrm{\ell }}^{\prime }(\eta ,\rho )\sim (\mathrm{\ell }+1)C_{\mathrm{\ell }}\left(\eta \right){\rho }^{\mathrm{\ell }}.$

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33.5.6 $$C_{\mathrm{\ell }}\left(0\right)=\frac{2^{\mathrm{\ell }}\mathrm{\ell }!}{(2\mathrm{\ell }+1)!}=\frac{1}{(2\mathrm{\ell }+1)!!}.$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $!!$: double factorial, $!$: factorial (as in $n!$) and $\mathrm{\ell }$: nonnegative integer

Permalink:

http://dlmf.nist.gov/33.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §33.5(ii), §33.5 and Ch.33

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§33.5(iv) Large $\mathrm{\ell }$

►As $\mathrm{\ell }\to \mathrm{\infty }$ with $\eta$ and $\rho$ ($\ne 0$) fixed, …

13: 33.14 Definitions and Basic Properties

… ►Again, there is a regular singularity at $r=0$ with indices $\mathrm{\ell }+1$ and $-\mathrm{\ell }$, and an irregular singularity of rank 1 at $r=\mathrm{\infty }$. … ►The functions $s(ϵ,\mathrm{\ell };r)$ and $c(ϵ,\mathrm{\ell };r)$ are defined by …An alternative formula for $A(ϵ,\mathrm{\ell })$ is … ►Note that the functions ${\varphi }_{n,\mathrm{\ell }}$, $n=\mathrm{\ell },\mathrm{\ell }+1,\mathrm{\dots }$, do not form a complete orthonormal system. … ►With arguments $ϵ,\mathrm{\ell },r$ suppressed, …

14: 33.8 Continued Fractions

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33.8.1 $$\frac{F_{\mathrm{\ell }}^{\prime }}{F_{\mathrm{\ell }}}=S_{\mathrm{\ell }+1}-\frac{R_{\mathrm{\ell }+1}^2}{T_{\mathrm{\ell }+1}-}\frac{R_{\mathrm{\ell }+2}^2}{T_{\mathrm{\ell }+2}-}\mathrm{\cdots }.$$

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Symbols:

$F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $R_{\mathrm{\ell }}$: factor, $S_{\mathrm{\ell }}$: factor and $T_{\mathrm{\ell }}$: factor

Referenced by:

§33.8, §33.8

Permalink:

http://dlmf.nist.gov/33.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.8 and Ch.33

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$a=1+\mathrm{\ell }\pm \mathrm{i}\eta ,$

… ►If we denote $u=F_{\mathrm{\ell }}^{\prime }/F_{\mathrm{\ell }}$ and $p+\mathrm{i}q=H_{\mathrm{\ell }}^{+}{}_{}{}^{\prime }/H_{\mathrm{\ell }}^{+}$, then … ►

$F_{\mathrm{\ell }}^{\prime }=u{F}_{\mathrm{\ell }},$

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$G_{\mathrm{\ell }}=q^{-1}(u-p)F_{\mathrm{\ell }},$

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15: 33.16 Connection Formulas

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§33.16(i) $F_{\mathrm{\ell }}$ and $G_{\mathrm{\ell }}$ in Terms of $f$ and $h$

… ►where $C_{\mathrm{\ell }}\left(\eta \right)$ is given by (33.2.5) or (33.2.6). … ►and again define $A(ϵ,\mathrm{\ell })$ by (33.14.11) or (33.14.12). … ►and again define $A(ϵ,\mathrm{\ell })$ by (33.14.11) or (33.14.12). … ►When denote $\nu$, ${\zeta }_{\mathrm{\ell }}(\nu ,r)$, and ${\xi }_{\mathrm{\ell }}(\nu ,r)$ by (33.16.8) and (33.16.9). …

16: 33.2 Definitions and Basic Properties

… ►This differential equation has a regular singularity at $\rho =0$ with indices $\mathrm{\ell }+1$ and $-\mathrm{\ell }$, and an irregular singularity of rank 1 at $\rho =\mathrm{\infty }$ (§§2.7(i), 2.7(ii)). … ►The normalizing constant $C_{\mathrm{\ell }}\left(\eta \right)$ is always positive, and has the alternative form … ► ${\sigma }_{\mathrm{\ell }}\left(\eta \right)$ is the Coulomb phase shift. … ► $H_{\mathrm{\ell }}^{+}(\eta ,\rho )$ and $H_{\mathrm{\ell }}^{-}(\eta ,\rho )$ are complex conjugates, and their real and imaginary parts are given by … ►As in the case of $F_{\mathrm{\ell }}(\eta ,\rho )$, the solutions $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ and $G_{\mathrm{\ell }}(\eta ,\rho )$ are analytic functions of $\rho$ when . …

17: 24.14 Sums

§24.14 Sums

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§24.14(i) Quadratic Recurrence Relations

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§24.14(ii) Higher-Order Recurrence Relations

►In the following two identities, valid for $n\ge 2$, the sums are taken over all nonnegative integers $j,k,\mathrm{\ell }$ with $j+k+\mathrm{\ell }=n$. … ►In the next identity, valid for $n\ge 4$, the sum is taken over all positive integers $j,k,\mathrm{\ell },m$ with $j+k+\mathrm{\ell }+m=n$. …

18: 33.19 Power-Series Expansions in $r$

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33.19.1 $$f(ϵ,\mathrm{\ell };r)=r^{\mathrm{\ell }+1}\sum _{k=0}^{\mathrm{\infty }}{\alpha }_k{r}^k,$$

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Symbols:

$f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter and ${\alpha }_k$: term

Referenced by:

§33.18, §33.19

Permalink:

http://dlmf.nist.gov/33.19.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.19 and Ch.33

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${\alpha }_0=2^{\mathrm{\ell }+1}/(2\mathrm{\ell }+1)!,$

… ►Here $\kappa$ is defined by (33.14.6), $A(ϵ,\mathrm{\ell })$ is defined by (33.14.11) or (33.14.12), ${\gamma }_0=1$, ${\gamma }_1=1$, and … ►

${\delta }_0=\left({\beta }_{2\mathrm{\ell }+1}-2(\psi \left(2\mathrm{\ell }+2\right)+\psi \left(1\right))A(ϵ,\mathrm{\ell })\right){\alpha }_0,$

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${\delta }_1=\left({\beta }_{2\mathrm{\ell }+2}-2(\psi \left(2\mathrm{\ell }+3\right)+\psi \left(2\right))A(ϵ,\mathrm{\ell })\right){\alpha }_1,$

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19: 33.20 Expansions for Small $|ϵ|$

… ►where … ►As $ϵ\to 0$ with $\mathrm{\ell }$ and $r$ fixed, …where $A(ϵ,\mathrm{\ell })$ is given by (33.14.11), (33.14.12), and … ►For a comprehensive collection of asymptotic expansions that cover $f(ϵ,\mathrm{\ell };r)$ and $h(ϵ,\mathrm{\ell };r)$ as $ϵ\to 0\pm$ and are uniform in $r$, including unbounded values, see Curtis (1964a, §7). These expansions are in terms of elementary functions, Airy functions, and Bessel functions of orders $2\mathrm{\ell }+1$ and $2\mathrm{\ell }+2$.

20: 18.5 Explicit Representations

… ►In (18.5.4_5) see §26.11 for the Fibonacci numbers $F_n$. … ►

18.5.7 $$P_n^{(\alpha ,\beta )}\left(x\right)=\sum _{\mathrm{\ell }=0}^n\frac{{\left(n+\alpha +\beta +1\right)}_{\mathrm{\ell }}{\left(\alpha +\mathrm{\ell }+1\right)}_{n-\mathrm{\ell }}}{\mathrm{\ell }!(n-\mathrm{\ell })!}{\left(\frac{x-1}{2}\right)}^{\mathrm{\ell }}=\frac{{\left(\alpha +1\right)}_n}{n!}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-n,n+\alpha +\beta +1}{\alpha +1};\frac{1-x}{2}),$$

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.42

Referenced by:

§18.17(iv), §18.17(vii), (18.27.12_5), Table 18.3.1, §18.5(iii), §18.5(iii), §18.6(ii), §18.9(iii)

Permalink:

http://dlmf.nist.gov/18.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

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18.5.11_1 $$T_n\left(x\right)=\frac{1}{2}n\sum _{\mathrm{\ell }=0}^{\lfloor n/2\rfloor }\frac{{(-1)}^{\mathrm{\ell }}(n-\mathrm{\ell }-1)!}{\mathrm{\ell }!(n-2\mathrm{\ell })!}{(2x)}^{n-2\mathrm{\ell }}=2^{n-1}{x}^n{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-\frac{1}{2}n,-\frac{1}{2}n+\frac{1}{2}}{1-n};\frac{1}{x^2}),$$ $n\ge 1$,

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Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $!$: factorial (as in $n!$), $\lfloor x\rfloor$: floor of $x$ , $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Source:

Erdélyi et al. (1953b, 10.11(22))

Referenced by:

§18.5(iii), Erratum (V1.2.0) Section 18.5

Permalink:

http://dlmf.nist.gov/18.5.E11_1

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

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18.5.11_3 $$U_n\left(x\right)=\sum _{\mathrm{\ell }=0}^{\lfloor n/2\rfloor }\frac{{(-1)}^{\mathrm{\ell }}(n-\mathrm{\ell })!}{\mathrm{\ell }!(n-2\mathrm{\ell })!}{(2x)}^{n-2\mathrm{\ell }}={\left(2x\right)}^n{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-\frac{1}{2}n,-\frac{1}{2}n+\frac{1}{2}}{-n};\frac{1}{x^2}),$$

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Symbols:

$U_n\left(x\right)$: Chebyshev polynomial of the second kind, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $!$: factorial (as in $n!$), $\lfloor x\rfloor$: floor of $x$ , $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Source:

Erdélyi et al. (1953b, 10.11(23))

Permalink:

http://dlmf.nist.gov/18.5.E11_3

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

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18.5.12 $$L_n^{(\alpha )}\left(x\right)=\sum _{\mathrm{\ell }=0}^n\frac{{\left(\alpha +\mathrm{\ell }+1\right)}_{n-\mathrm{\ell }}}{(n-\mathrm{\ell })!\mathrm{\ell }!}{(-x)}^{\mathrm{\ell }}=\frac{{\left(\alpha +1\right)}_n}{n!}{}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-n}{\alpha +1};x).$$

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Symbols:

${}_1{}^{}F_1^{}(a;b;z)$: $=M(a,b,z)$ notation for the Kummer confluent hypergeometric function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.54

Referenced by:

§18.11(i), §18.17(v), §18.17(vi), §18.17(vii), §18.18(iii), §18.21(ii), Table 18.3.1, (18.34.1), §18.34(i), §18.5(iii), §18.6(ii)

Permalink:

http://dlmf.nist.gov/18.5.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

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