Euler numbers

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11: 17.3 $q$-Elementary and $q$-Special Functions

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§17.3(iii) Bernoulli Polynomials; Euler and Stirling Numbers

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$q$-Euler Numbers

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17.3.8 $$A_{m,s}\left(q\right)=q^{\left(\genfrac{}{}{0.0pt}{}{s-m}{2}\right)+\left(\genfrac{}{}{0.0pt}{}{s}{2}\right)}\sum _{j=0}^s{(-1)}^j{q}^{\left(\genfrac{}{}{0.0pt}{}{j}{2}\right)}{\left[\genfrac{}{}{0.0pt}{}{m+1}{j}\right]}_q\frac{{(1-q^{s-j})}^m}{{(1-q)}^m}.$$

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Defines:

$A_{m,s}\left(q\right)$: $q$-Euler number

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $q$: complex base, $j$: nonnegative integer, $m$: nonnegative integer and $s$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.3.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §17.3(iii), §17.3(iii), §17.3 and Ch.17

… ►The $A_{m,s}\left(q\right)$ are always polynomials in $q$, and the $a_{m,s}\left(q\right)$ are polynomials in $q$ for $0\le s\le m$. …

12: 24.19 Methods of Computation

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§24.19(i) Bernoulli and Euler Numbers and Polynomials

►Equations (24.5.3) and (24.5.4) enable $B_n$ and $E_n$ to be computed by recurrence. …A similar method can be used for the Euler numbers based on (4.19.5). … ►For algorithms for computing $B_n$, $E_n$, $B_n\left(x\right)$, and $E_n\left(x\right)$ see Spanier and Oldham (1987, pp. 37, 41, 171, and 179–180). …

13: 24.4 Basic Properties

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§24.4(iv) Finite Expansions

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24.4.16 $$E_{2n}=\frac{1}{2n+1}-\sum _{k=1}^n\left(\genfrac{}{}{0pt}{}{2n}{2k-1}\right)\frac{2^{2k}(2^{2k-1}-1)B_{2k}}{k},$$

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Symbols:

$B_n$: Bernoulli numbers, $E_n$: Euler numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer and $n$: integer

Permalink:

http://dlmf.nist.gov/24.4.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §24.4(iv), §24.4 and Ch.24

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24.4.17 $$E_{2n}=1-\sum _{k=1}^n\left(\genfrac{}{}{0pt}{}{2n}{2k-1}\right)\frac{2^{2k}(2^{2k}-1)B_{2k}}{2k}.$$

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Symbols:

$B_n$: Bernoulli numbers, $E_n$: Euler numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer and $n$: integer

Permalink:

http://dlmf.nist.gov/24.4.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §24.4(iv), §24.4 and Ch.24

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24.4.28 $$E_n\left(\frac{1}{2}\right)=2^{-n}{E}_n.$$

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Symbols:

$E_n$: Euler numbers, $E_n\left(x\right)$: Euler polynomials and $n$: integer

A&S Ref:

23.1.21

Referenced by:

§24.9

Permalink:

http://dlmf.nist.gov/24.4.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §24.4(vi), §24.4 and Ch.24

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24.4.33 $$E_{2n}\left(\frac{1}{6}\right)=E_{2n}\left(\frac{5}{6}\right)=\frac{1+3^{-2n}}{2^{2n+1}}{E}_{2n}.$$

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Symbols:

$E_n$: Euler numbers, $E_n\left(x\right)$: Euler polynomials and $n$: integer

Referenced by:

§24.4(vi)

Permalink:

http://dlmf.nist.gov/24.4.E33

Encodings:

pMML, png, TeX

See also:

Annotations for §24.4(vi), §24.4 and Ch.24

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14: 24.13 Integrals

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24.13.5 $${\int }_{1/4}^{3/4}{B}_n\left(t\right)dt=\frac{E_n}{2^{2n+1}}.$$

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $E_n$: Euler numbers, $dx$: 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

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24.13.8 $${\int }_0^1{E}_n\left(t\right)dt=-2\frac{E_{n+1}\left(0\right)}{n+1}=\frac{4(2^{n+2}-1)}{(n+1)(n+2)}{B}_{n+2},$$

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Symbols:

$B_n$: Bernoulli numbers, $E_n\left(x\right)$: Euler polynomials, $dx$: 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

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24.13.9 $${\int }_0^{1/2}{E}_{2n}\left(t\right)dt=-\frac{E_{2n+1}\left(0\right)}{2n+1}=\frac{2(2^{2n+2}-1)B_{2n+2}}{(2n+1)(2n+2)},$$

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Symbols:

$B_n$: Bernoulli numbers, $E_n\left(x\right)$: Euler polynomials, $dx$: differential of $x$ , $\int$: integral, $n$: integer and $t$: real or complex

Permalink:

http://dlmf.nist.gov/24.13.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §24.13(ii), §24.13 and Ch.24

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24.13.10 $${\int }_0^{1/2}{E}_{2n-1}\left(t\right)dt=\frac{E_{2n}}{n{2}^{2n+1}},$$ $n=1,2,\mathrm{\dots }$.

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Symbols:

$E_n$: Euler numbers, $E_n\left(x\right)$: Euler polynomials, $dx$: 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

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24.13.11 $${\int }_0^1{E}_n\left(t\right)E_m\left(t\right)dt={(-1)}^{n}4\frac{(2^{m+n+2}-1)m!n!}{(m+n+2)!}{B}_{m+n+2}.$$

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Symbols:

$B_n$: Bernoulli numbers, $E_n\left(x\right)$: Euler polynomials, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $m$: integer, $n$: integer and $t$: real or complex

A&S Ref:

23.1.12

Referenced by:

§24.13(ii)

Permalink:

http://dlmf.nist.gov/24.13.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §24.13(ii), §24.13 and Ch.24

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15: 4.19 Maclaurin Series and Laurent Series

… ►In (4.19.3)–(4.19.9), $B_n$ are the Bernoulli numbers and $E_n$ are the Euler numbers (§§24.2(i)–24.2(ii)). … ►

4.19.5 $$\sec z=1+\frac{z^2}{2}+\frac{5}{24}{z}^4+\frac{61}{720}{z}^6+\mathrm{\cdots }+\frac{{(-1)}^n{E}_{2n}}{(2n)!}{z}^{2n}+\mathrm{\cdots },$$ ,

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Symbols:

$E_n$: Euler numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\sec z$: secant function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.69

Referenced by:

§24.1, §24.19(i), §24.2(ii)

Permalink:

http://dlmf.nist.gov/4.19.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.19 and Ch.4

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16: 24.7 Integral Representations

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§24.7(i) Bernoulli and Euler Numbers

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24.7.5 $$B_{2n}={(-1)}^n\frac{2n(2n-1)}{\pi }{\int }_0^{\mathrm{\infty }}{t}^{2n-2}\ln \left(1-{\mathrm{e}}^{-2\pi t}\right)dt.$$

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Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\ln z$: principal branch of logarithm function, $n$: integer and $t$: real or complex

Referenced by:

§24.7(i)

Permalink:

http://dlmf.nist.gov/24.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §24.7(i), §24.7 and Ch.24

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24.7.6 $$E_{2n}={(-1)}^n{2}^{2n+1}{\int }_0^{\mathrm{\infty }}{t}^{2n}\mathrm{sech}\left(\pi t\right)dt.$$

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Symbols:

$E_n$: Euler numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{sech}z$: hyperbolic secant function, $\int$: integral, $n$: integer and $t$: real or complex

Referenced by:

§24.7(i)

Permalink:

http://dlmf.nist.gov/24.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §24.7(i), §24.7 and Ch.24

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17: 24.8 Series Expansions

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24.8.9 $$E_{2n}={(-1)}^n\sum _{k=1}^{\mathrm{\infty }}\frac{k^{2n}}{\cosh \left(\frac{1}{2}\pi k\right)}-4\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{(2k+1)}^{2n}}{{\mathrm{e}}^{2\pi (2k+1)}-1},$$ $n=1,2,\mathrm{\dots }$.

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Symbols:

$E_n$: Euler numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $k$: integer and $n$: integer

Keywords:

Euler polynomials, infinite series expansions, other

Permalink:

http://dlmf.nist.gov/24.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §24.8(ii), §24.8 and Ch.24

18: 24.15 Related Sequences of Numbers

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24.15.12 $$\sum _{k=0}^{\lfloor n/2\rfloor }\left(\genfrac{}{}{0pt}{}{n}{2k}\right){\left(\frac{5}{4}\right)}^k{E}_{2k}{v}_{n-2k}=\frac{1}{2^{n-1}}.$$

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Symbols:

$E_n$: Euler numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\lfloor x\rfloor$: floor of $x$ , $k$: integer, $n$: integer and $v_n$: Lucas numbers

Permalink:

http://dlmf.nist.gov/24.15.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §24.15(iv), §24.15 and Ch.24

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19: 24.17 Mathematical Applications

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§24.17(iii) Number Theory

►Bernoulli and Euler numbers and polynomials occur in: number theory via (24.4.7), (24.4.8), and other identities involving sums of powers; the Riemann zeta function and $L$-series (§25.15, Apostol (1976), and Ireland and Rosen (1990)); arithmetic of cyclotomic fields and the classical theory of Fermat’s last theorem (Ribenboim (1979) and Washington (1997)); $p$-adic analysis (Koblitz (1984, Chapter 2)). …

20: 24.16 Generalizations

§24.16 Generalizations

… ►When $x=0$ they reduce to the Bernoulli and Euler numbers of order $\mathrm{\ell }$ : … ►

24.16.13 $$E_n\left(x\right)=-\frac{2^{1-n}}{n+1}{B}_{n+1,{\chi }_4}(2x-1).$$

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Symbols:

$E_n\left(x\right)$: Euler polynomials, $n$: integer, $x$: real or complex, $\chi (a)$: Dirichlet character and $B_{n,\chi }$: Generalized Bernoulli numbers

Referenced by:

§24.16(ii)

Permalink:

http://dlmf.nist.gov/24.16.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §24.16(ii), §24.16 and Ch.24

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§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)).