Euler pentagonal number theorem

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21: 27.15 Chinese Remainder Theorem

§27.15 Chinese Remainder Theorem

… ►This theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation. …Their product

m

has 20 digits, twice the number of digits in the data. By the Chinese remainder theorem each integer in the data can be uniquely represented by its residues (mod

m_{1}

), (mod

m_{2}

), (mod

m_{3}

), and (mod

m_{4}

), respectively. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result

\pmod{m}

, which is correct to 20 digits. …

23: 24.13 Integrals

… ►

24.13.5

\int_{1/4}^{3/4}B_{n}\left(t\right)\,\mathrm{d}t=\frac{E_{n}}{2^{2n+1}}.

ⓘ

Symbols:: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $E_{\NVar{n}}$ : Euler numbers, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $n$ : integer and $t$ : real or complex
Referenced by:: §24.13(i)
Permalink:: http://dlmf.nist.gov/24.13.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §24.13(i), §24.13 and Ch.24

… ►

§24.13(ii) Euler Polynomials

… ►

24.13.8

\int_{0}^{1}E_{n}\left(t\right)\,\mathrm{d}t=-2\frac{E_{n+1}\left(0\right)}{n+% 1}=\frac{4(2^{n+2}-1)}{(n+1)(n+2)}B_{n+2},

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $n$ : integer and $t$ : real or complex
Permalink:: http://dlmf.nist.gov/24.13.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §24.13(ii), §24.13 and Ch.24

… ►

24.13.10

\int_{0}^{1/2}E_{2n-1}\left(t\right)\,\mathrm{d}t=\frac{E_{2n}}{n2^{2n+1}},

n=1,2,\dots

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $n$ : integer and $t$ : real or complex
Referenced by:: §24.13(ii)
Permalink:: http://dlmf.nist.gov/24.13.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §24.13(ii), §24.13 and Ch.24

… ►

§24.13(iii) Compendia

…

24: 24.9 Inequalities

§24.9 Inequalities

… ►

24.9.3

4^{-n}|E_{2n}|>(-1)^{n}E_{2n}\left(x\right)>0,

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $n$ : integer and $x$ : real or complex
A&S Ref:: 23.1.13
Referenced by:: §24.9, §24.9
Permalink:: http://dlmf.nist.gov/24.9.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §24.9 and Ch.24

… ►

24.9.5

\frac{4(2n-1)!}{\pi^{2n}}\frac{2^{2n}-1}{2^{2n}-2}>(-1)^{n}E_{2n-1}\left(x% \right)>0.

ⓘ

Symbols:: $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $n$ : integer and $x$ : real or complex
A&S Ref:: 23.1.14
Referenced by:: §24.9, §24.9
Permalink:: http://dlmf.nist.gov/24.9.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §24.9 and Ch.24

… ►

24.9.7

8\sqrt{\frac{n}{\pi}}\left(\frac{4n}{\pi e}\right)^{2n}\left(1+\frac{1}{12n}% \right)>(-1)^{n}E_{2n}>8\sqrt{\frac{n}{\pi}}\left(\frac{4n}{\pi e}\right)^{2n}.

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm and $n$ : integer
Referenced by:: §24.9, §24.9
Permalink:: http://dlmf.nist.gov/24.9.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §24.9 and Ch.24

… ►

24.9.10

\frac{4^{n+1}(2n)!}{\pi^{2n+1}}>(-1)^{n}E_{2n}>\frac{4^{n+1}(2n)!}{\pi^{2n+1}}% \frac{1}{1+3^{-1-2n}}.

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ) and $n$ : integer
A&S Ref:: 23.1.15
Referenced by:: §24.9
Permalink:: http://dlmf.nist.gov/24.9.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §24.9 and Ch.24

25: 30.10 Series and Integrals

… ►Integrals and integral equations for

\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)

are given in Arscott (1964b, §8.6), Erdélyi et al. (1955, §16.13), Flammer (1957, Chapter 5), and Meixner (1951). …For an addition theorem, see Meixner and Schäfke (1954, p. 300) and King and Van Buren (1973). …

26: 24.16 Generalizations

§24.16 Generalizations

… ►For

\ell=0,1,2,\dotsc

, Bernoulli and Euler polynomials of order $\ell$ are defined respectively by …When

x=0

they reduce to the Bernoulli and Euler numbers of order $\ell$ : … ►

§24.16(iii) Other Generalizations

►In no particular order, other generalizations include: Bernoulli numbers and polynomials with arbitrary complex index (Butzer et al. (1992)); Euler numbers and polynomials with arbitrary complex index (Butzer et al. (1994)); q-analogs (Carlitz (1954a), Andrews and Foata (1980)); conjugate Bernoulli and Euler polynomials (Hauss (1997, 1998)); Bernoulli–Hurwitz numbers (Katz (1975)); poly-Bernoulli numbers (Kaneko (1997)); Universal Bernoulli numbers (Clarke (1989));

p

-adic integer order Bernoulli numbers (Adelberg (1996));

p

-adic

q

-Bernoulli numbers (Kim and Kim (1999)); periodic Bernoulli numbers (Berndt (1975b)); cotangent numbers (Girstmair (1990b)); Bernoulli–Carlitz numbers (Goss (1978)); Bernoulli–Padé numbers (Dilcher (2002)); Bernoulli numbers belonging to periodic functions (Urbanowicz (1988)); cyclotomic Bernoulli numbers (Girstmair (1990a)); modified Bernoulli numbers (Zagier (1998)); higher-order Bernoulli and Euler polynomials with multiple parameters (Erdélyi et al. (1953a, §§1.13.1, 1.14.1)).

27: 27.3 Multiplicative Properties

§27.3 Multiplicative Properties

►Except for

\nu\left(n\right)

\Lambda\left(n\right)

p_{n}

, and

\pi\left(x\right)

, the functions in §27.2 are multiplicative, which means

f(1)=1

and … ►

27.3.2

f(n)=\prod_{r=1}^{\nu\left(n\right)}f(p^{a_{r}}_{r}).

ⓘ

Symbols:: $\nu\left(\NVar{n}\right)$ : number of distinct primes dividing a number, $n$ : positive integer, $p,p_{1},\ldots$ : prime numbers, $a_{r}$ : positive exponent and $f(n)$ : multiplicative function
Referenced by:: §27.20, §27.20, §27.3
Permalink:: http://dlmf.nist.gov/27.3.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §27.3 and Ch.27

… ►

27.3.3

\phi\left(n\right)=n\prod_{p\mathbin{|}n}(1-p^{-1}),

ⓘ

Symbols:: $\phi\left(\NVar{n}\right)$ : Euler’s totient, $n$ : positive integer and $p,p_{1},\ldots$ : prime numbers
A&S Ref:: 24.3.2 I.C
Permalink:: http://dlmf.nist.gov/27.3.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §27.3 and Ch.27

… ►

27.3.8

\phi\left(m\right)\phi\left(n\right)=\phi\left(mn\right)\phi\left(\left(m,n% \right)\right)/\left(m,n\right).

ⓘ

Symbols:: $\phi\left(\NVar{n}\right)$ : Euler’s totient, $\left(\NVar{m},\NVar{n}\right)$ : greatest common divisor (gcd), $m$ : positive integer and $n$ : positive integer
Permalink:: http://dlmf.nist.gov/27.3.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §27.3 and Ch.27

…

28: 24.20 Tables

§24.20 Tables

… ►Wagstaff (1978) gives complete prime factorizations of

N_{n}

and

E_{n}

for

n=20(2)60

and

n=8(2)42

, respectively. …

29: 24.11 Asymptotic Approximations

§24.11 Asymptotic Approximations

… ►

24.11.1

(-1)^{n+1}B_{2n}\sim\frac{2(2n)!}{(2\pi)^{2n}},

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ) and $n$ : integer
Referenced by:: §24.11
Permalink:: http://dlmf.nist.gov/24.11.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §24.11 and Ch.24

… ►

24.11.3

(-1)^{n}E_{2n}\sim\frac{2^{2n+2}(2n)!}{\pi^{2n+1}},

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ) and $n$ : integer
Permalink:: http://dlmf.nist.gov/24.11.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §24.11 and Ch.24

►

24.11.4

(-1)^{n}E_{2n}\sim 8\sqrt{\frac{n}{\pi}}\left(\frac{4n}{\pi e}\right)^{2n}.

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm and $n$ : integer
Referenced by:: §24.11
Permalink:: http://dlmf.nist.gov/24.11.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §24.11 and Ch.24

… ►

24.11.6

(-1)^{\left\lfloor(n+1)/2\right\rfloor}\frac{\pi^{n+1}}{4(n!)}E_{n}\left(x% \right)\to\begin{cases}\sin\left(\pi x\right),&n\text{ even},\\ \cos\left(\pi x\right),&n\text{ odd},\end{cases}

ⓘ

Symbols:: $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $!$ : factorial (as in $n!$ ), $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $\sin\NVar{z}$ : 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

…

30: 5.2 Definitions

… ►

Euler’s Integral

… ►When

\Re z\leq 0

\Gamma\left(z\right)

is defined by analytic continuation. …

1/\Gamma\left(z\right)

is entire, with simple zeros at

z=-n

. … ►

§5.2(ii) Euler’s Constant

… ►