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1: 24.1 Special Notation

… ►Unless otherwise noted, the formulas in this chapter hold for all values of the variables $x$ and $t$, and for all nonnegative integers $n$. ►

Bernoulli Numbers and Polynomials

►The origin of the notation $B_n$, $B_n\left(x\right)$, is not clear. … ►

Euler Numbers and Polynomials

… ►Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. …

2: 24.10 Arithmetic Properties

… ►The denominator of $B_{2n}$ is the product of all these primes $p$. …where $n\ge 2$, and $\mathrm{\ell }(\ge 1)$ is an arbitrary integer such that $(p-1)p^{\mathrm{\ell }}|2n$. … ►valid for fixed integers $\mathrm{\ell }(\ge 0)$, and for all $n(\ge 0)$ and $w(\ge 0)$ such that $2^{\mathrm{\ell }}|w$. … ►valid for fixed integers $\mathrm{\ell }(\ge 1)$, and for all $n(\ge 1)$ such that $2n\cancel{\equiv }0$ $(modp-1)$ and $p^{\mathrm{\ell }}|2n$. …valid for fixed integers $\mathrm{\ell }(\ge 1)$ and for all $n(\ge 1)$ such that $(p-1)p^{\mathrm{\ell }-1}|2n$.

3: 33.6 Power-Series Expansions in $\rho$

… ►where $A_{\mathrm{\ell }+1}^{\mathrm{\ell }}=1$, $A_{\mathrm{\ell }+2}^{\mathrm{\ell }}=\eta /(\mathrm{\ell }+1)$, and ►

33.6.3 $$(k+\mathrm{\ell })(k-\mathrm{\ell }-1)A_k^{\mathrm{\ell }}=2\eta A_{k-1}^{\mathrm{\ell }}-A_{k-2}^{\mathrm{\ell }},$$ $k=\mathrm{\ell }+3,\mathrm{\ell }+4,\mathrm{\dots }$,

ⓘ

Symbols:

$k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\eta$: real parameter and $A_{\mathrm{\ell }+1}^{\mathrm{\ell }}$: coefficient

A&S Ref:

14.1.6

Permalink:

http://dlmf.nist.gov/33.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §33.6 and Ch.33

… ►where $a=1+\mathrm{\ell }\pm \mathrm{i}\eta$ and $\psi \left(x\right)={\mathrm{\Gamma }}^{\prime }\left(x\right)/\mathrm{\Gamma }\left(x\right)$ (§5.2(i)). ►The series (33.6.1), (33.6.2), and (33.6.5) converge for all finite values of $\rho$. Corresponding expansions for $H_{\mathrm{\ell }}^{\pm }{}_{}{}^{\prime }(\eta ,\rho )$ can be obtained by combining (33.6.5) with (33.4.3) or (33.4.4).

4: 33.8 Continued Fractions

… ►

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

ⓘ

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

… ►The continued fraction (33.8.1) converges for all finite values of $\rho$, and (33.8.2) converges for all $\rho \ne 0$. ►If we denote $u=F_{\mathrm{\ell }}^{\prime }/F_{\mathrm{\ell }}$ and $p+\mathrm{i}q=H_{\mathrm{\ell }}^{+}{}_{}{}^{\prime }/H_{\mathrm{\ell }}^{+}$, then … ►

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

►

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

…

5: 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 . …

6: 33.9 Expansions in Series of Bessel Functions

… ►where the function $𝗃$ is as in §10.47(ii), $a_{-1}=0$, $a_0=(2\mathrm{\ell }+1)!!C_{\mathrm{\ell }}\left(\eta \right)$, and …The series (33.9.1) converges for all finite values of $\eta$ and $\rho$. … ►Here $b_{2\mathrm{\ell }}=b_{2\mathrm{\ell }+2}=0$, $b_{2\mathrm{\ell }+1}=1$, and …The series (33.9.3) and (33.9.4) converge for all finite positive values of $\left|\eta \right|$ and $\rho$. … ►For other asymptotic expansions of $G_{\mathrm{\ell }}(\eta ,\rho )$ see Fröberg (1955, §8) and Humblet (1985).

7: 33.19 Power-Series Expansions in $r$

… ►

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

… ►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 … ►

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

►

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

… ►The expansions (33.19.1) and (33.19.3) converge for all finite values of $r$, except $r=0$ in the case of (33.19.3).

8: 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 $2\mathrm{\ell }+2$.

9: 24.14 Sums

§24.14 Sums

►

§24.14(i) Quadratic Recurrence Relations

… ►

§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 $j+k+\mathrm{\ell }+m=n$. …

10: 33.23 Methods of Computation

… ►In a similar manner to §33.23(iii) the recurrence relations of §§33.4 or 33.17 can be used for a range of values of the integer $\mathrm{\ell }$, provided that the recurrence is carried out in a stable direction (§3.6). This implies decreasing $\mathrm{\ell }$ for the regular solutions and increasing $\mathrm{\ell }$ for the irregular solutions of §§33.2(iii) and 33.14(iii). … ►§33.8 supplies continued fractions for $F_{\mathrm{\ell }}^{\prime }/F_{\mathrm{\ell }}$ and $H_{\mathrm{\ell }}^{\pm }{}_{}{}^{\prime }/H_{\mathrm{\ell }}^{\pm }$. Combined with the Wronskians (33.2.12), the values of $F_{\mathrm{\ell }}$, $G_{\mathrm{\ell }}$, and their derivatives can be extracted. … ►Bardin et al. (1972) describes ten different methods for the calculation of $F_{\mathrm{\ell }}$ and $G_{\mathrm{\ell }}$, valid in different regions of the ($\eta ,\rho$)-plane. …