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1: 10.59 Integrals

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2: 23.18 Modular Transformations

… ►Here e and o are generic symbols for even and odd integers, respectively. In particular, if $a-1,b,c$, and $d-1$ are all even, then …

3: 27.13 Functions

… ►Every even integer $n>4$ is the sum of two odd primes. In this case, $S(n)$ is the number of solutions of the equation $n=p+q$, where $p$ and $q$ are odd primes. Goldbach’s assertion is that $S(n)\ge 1$ for all even $n>4$. …Vinogradov (1937) proves that every sufficiently large odd integer is the sum of three odd primes, and Chen (1966) shows that every sufficiently large even integer is the sum of a prime and a number with no more than two prime factors. … ►By similar methods Jacobi proved that $r_4\left(n\right)=8{\sigma }_1\left(n\right)$ if $n$ is odd, whereas, if $n$ is even, $r_4\left(n\right)=24$ times the sum of the odd divisors of $n$. …Exact formulas for $r_k\left(n\right)$ have also been found for $k=3,5$, and $7$, and for all even $k\le 24$. …

4: 29.12 Definitions

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Table 29.12.1: Lamé polynomials.

► ►►►►►►►

$\nu$	eigenvalue $h$	eigenfunction $w(z)$	polynomial form	real period	imag. period	parity of $w(z)$	parity of $w(z-K)$	parity of $w(z-K-\mathrm{i}{K}^{\prime })$
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$2n+1$	$a_{\nu }^{2m+1}\left(k^2\right)$	${\mathrm{𝑠𝐸}}_{\nu }^m(z,k^2)$	$\mathrm{sn}P({\mathrm{sn}}^2)$	$4K$	$2\mathrm{i}{K}^{\prime }$	odd	even	even
$2n+1$	$b_{\nu }^{2m+1}\left(k^2\right)$	${\mathrm{𝑐𝐸}}_{\nu }^m(z,k^2)$	$\mathrm{cn}P({\mathrm{sn}}^2)$	$4K$	$4\mathrm{i}{K}^{\prime }$	even	odd	even
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$2n+2$	$b_{\nu }^{2m+2}\left(k^2\right)$	${\mathrm{𝑠𝑐𝐸}}_{\nu }^m(z,k^2)$	$\mathrm{sn}\mathrm{cn}P({\mathrm{sn}}^2)$	$2K$	$4\mathrm{i}{K}^{\prime }$	odd	odd	even
$2n+2$	$a_{\nu }^{2m+1}\left(k^2\right)$	${\mathrm{𝑠𝑑𝐸}}_{\nu }^m(z,k^2)$	$\mathrm{sn}\mathrm{dn}P({\mathrm{sn}}^2)$	$4K$	$4\mathrm{i}{K}^{\prime }$	odd	even	odd
$2n+2$	$b_{\nu }^{2m+1}\left(k^2\right)$	${\mathrm{𝑐𝑑𝐸}}_{\nu }^m(z,k^2)$	$\mathrm{cn}\mathrm{dn}P({\mathrm{sn}}^2)$	$4K$	$2\mathrm{i}{K}^{\prime }$	even	odd	odd
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5: 26.13 Permutations: Cycle Notation

… ►For the example (26.13.2), this decomposition is given by $\left(1,3,2,5,7\right)\left(6,8\right)=\left(1,3\right)\left(2,3\right)\left(2,5\right)\left(5,7\right)\left(6,8\right).$ ►A permutation is even or odd according to the parity of the number of transpositions. The sign of a permutation is $+$ if the permutation is even, $-$ if it is odd. …

6: 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

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7: 14.16 Zeros

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$\mu >0$, , and $m-n$ is odd.

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$\mu >0$, , and $m-n$ is even.

… ► ${𝖰}_{\nu }^{\mu }\left(x\right)$ has $\max (\lceil \nu -|\mu |\rceil ,0)+k$ zeros in the interval $(-1,1)$, where $k$ can take one of the values $-1$, $0$, $1$, $2$, subject to $\max (\lceil \nu -|\mu |\rceil ,0)+k$ being even or odd according as $\cos \left(\nu \pi \right)$ and $\cos \left(\mu \pi \right)$ have opposite signs or the same sign. … ►

$\mu \le \nu$, $\mu \notin \mathbb{Z}$, and $\lfloor \mu \rfloor$ is odd.

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8: 24.12 Zeros

… ►When $n(\ge 2)$ is even … ►When $n$ is odd $x_1^{(n)}=\frac{1}{2}$, $x_2^{(n)}=1$ $(n\ge 3)$, and as $n\to \mathrm{\infty }$ with $m(\ge 1)$ fixed, … ►When $n(\ge 2)$ is even $y_1^{(n)}=1$, and as $n\to \mathrm{\infty }$ with $m(\ge 1)$ fixed, … ►When $n$ is odd $y_1^{(n)}=\frac{1}{2}$, … ►The only polynomial $E_n\left(x\right)$ with multiple zeros is $E_5\left(x\right)=(x-\frac{1}{2}){(x^2-x-1)}^2$.

9: 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. …

10: 30.16 Methods of Computation

… ►Let $n-m$ be even. … ►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. …