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11: 29.17 Other Solutions

… ►Algebraic Lamé functions are solutions of (29.2.1) when $\nu$ is half an odd integer. …

12: 24.11 Asymptotic Approximations

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24.11.5

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $!$: factorial (as in $n!$), $\lfloor x\rfloor$: floor of $x$ , $\sin z$: sine function, $n$: integer and $x$: real or complex

Referenced by:

§24.11

Permalink:

http://dlmf.nist.gov/24.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §24.11 and Ch.24

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24.11.6

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Symbols:

$E_n\left(x\right)$: Euler polynomials, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $!$: factorial (as in $n!$), $\lfloor x\rfloor$: floor of $x$ , $\sin z$: sine function, $n$: integer and $x$: real or complex

Referenced by:

§24.11

Permalink:

http://dlmf.nist.gov/24.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §24.11 and Ch.24

…

13: 30.4 Functions of the First Kind

… ►the sign of ${\mathrm{𝖯𝗌}}_n^m(0,{\gamma }^2)$ being ${(-1)}^{(n+m)/2}$ when $n-m$ is even, and the sign of ${d{\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)/dx|}_{x=0}$ being ${(-1)}^{(n+m-1)/2}$ when $n-m$ is odd. … ►with ${\alpha }_k$, ${\beta }_k$, ${\gamma }_k$ from (30.3.6), and $g_{-1}=g_{-2}=0$, $g_k=0$ for even $k$ if $n-m$ is odd and $g_k=0$ for odd $k$ if $n-m$ is even. …

14: 30.16 Methods of Computation

… ►If $n-m$ is odd, then (30.16.1) is replaced by … ►If ${\lambda }_n^m\left({\gamma }^2\right)$ is known, then we can compute ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ (not normalized) by solving the differential equation (30.2.1) numerically with initial conditions $w(0)=1$, $w^{\prime }(0)=0$ if $n-m$ is even, or $w(0)=0$, $w^{\prime }(0)=1$ if $n-m$ is odd. … ►Let $𝐀$ be the $d\times d$ matrix given by (30.16.1) if $n-m$ is even, or by (30.16.6) if $n-m$ is odd. …

15: 12.4 Power-Series Expansions

… ►where the initial values are given by (12.2.6)–(12.2.9), and $u_1(a,z)$ and $u_2(a,z)$ are the even and odd solutions of (12.2.2) given by …

16: 29.3 Definitions and Basic Properties

… ►For each pair of values of $\nu$ and $k$ there are four infinite unbounded sets of real eigenvalues $h$ for which equation (29.2.1) has even or odd solutions with periods $2K$ or $4K$. … ►

Table 29.3.1: Eigenvalues of Lamé’s equation.

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eigenvalue $h$	parity	period
…
$a_{\nu }^{2m+1}\left(k^2\right)$	odd	$4K$
…
$b_{\nu }^{2m+2}\left(k^2\right)$	odd	$2K$

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… ►

Table 29.3.2: Lamé functions.

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boundary conditions	eigenvalue $h$	eigenfunction $w(z)$	parity of $w(z)$	parity of $w(z-K)$	period of $w(z)$
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$w(0)={dw/dz|}_{z=K}=0$	$a_{\nu }^{2m+1}\left(k^2\right)$	${\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(z,k^2)$	odd	even	$4K$
…
$w(0)=w(K)=0$	$b_{\nu }^{2m+2}\left(k^2\right)$	${\mathrm{𝐸𝑠}}_{\nu }^{2m+2}(z,k^2)$	odd	odd	$2K$

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17: 1.12 Continued Fractions

… ►If $C_n^{\prime }=C_{2n+1}$, $n=0,1,2,\mathrm{\dots }$, then $C^{\prime }$ is called the odd part of $C$. The odd part of $C$ exists iff $b_{2k+1}\ne 0$, $k=0,1,2,\mathrm{\dots }$, and up to equivalence is given by … ►and the even and odd parts of the continued fraction converge to finite values. …

18: 28.3 Graphics

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Odd $\pi$-Antiperiodic Solutions

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Odd $\pi$-Periodic Solutions

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19: 26.10 Integer Partitions: Other Restrictions

… ► $p(𝒪,n)$ denotes the number of partitions of $n$ into odd parts. … ►

26.10.6 $$p(𝒟,n)=\frac{1}{n}\sum _{t=1}^{n}p(𝒟,n-t)\sum _{\begin{array}{c}j|t\\ j\text{ <span class="ltx\_text hilite">odd</span>}\end{array}}j,$$

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Symbols:

$p(\mathrm{condition},n)$: restricted number of partitions of $n$ , $j$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.10.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §26.10(iii), §26.10 and Ch.26

►where the inner sum is the sum of all positive odd divisors of $t$. …

20: 28.5 Second Solutions ${\mathrm{fe}}_n$, ${\mathrm{ge}}_n$

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28.5.3

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Defines:

$f_n(z,q)$: functions (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $m$: integer, $q=h^2$: parameter, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/28.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.5(i), §28.5(i), §28.5 and Ch.28

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Odd Second Solutions

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