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1: 24.6 Explicit Formulas

§24.6 Explicit Formulas

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24.6.6 $$E_{2n}=\sum _{k=1}^{2n}\frac{{(-1)}^k}{2^{k-1}}\left(\genfrac{}{}{0pt}{}{2n+1}{k+1}\right)\sum _{j=0}^{\lfloor \frac{1}{2}k-\frac{1}{2}\rfloor }\left(\genfrac{}{}{0pt}{}{k}{j}\right){(k-2j)}^{2n}.$$

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Symbols:

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

Referenced by:

§24.6

Permalink:

http://dlmf.nist.gov/24.6.E6

Encodings:

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See also:

Annotations for §24.6 and Ch.24

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24.6.7 $$B_n\left(x\right)=\sum _{k=0}^n\frac{1}{k+1}\sum _{j=0}^k{(-1)}^j\left(\genfrac{}{}{0pt}{}{k}{j}\right){(x+j)}^n,$$

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $j$: integer, $k$: integer, $n$: integer and $x$: real or complex

Referenced by:

§24.6, §24.6

Permalink:

http://dlmf.nist.gov/24.6.E7

Encodings:

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See also:

Annotations for §24.6 and Ch.24

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24.6.12 $$E_{2n}=\sum _{k=0}^{2n}\frac{1}{2^k}\sum _{j=0}^k{(-1)}^j\left(\genfrac{}{}{0pt}{}{k}{j}\right){(1+2j)}^{2n}.$$

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Symbols:

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

Referenced by:

§24.6, §24.6

Permalink:

http://dlmf.nist.gov/24.6.E12

Encodings:

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See also:

Annotations for §24.6 and Ch.24

2: 10.49 Explicit Formulas

§10.49 Explicit Formulas

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§10.49(i) Unmodified Functions

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10.49.7 $${𝗁}_n^{(2)}\left(z\right)={\mathrm{e}}^{-\mathrm{i}z}\sum _{k=0}^n{(-\mathrm{i})}^{k-n-1}\frac{a_k(n+\frac{1}{2})}{z^{k+1}}.$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${𝗁}_n^{(2)}\left(z\right)$: spherical Bessel function of the third kind, $n$: integer, $k$: nonnegative integer, $z$: complex variable and $a_k(\nu )$: polynomial coefficient

A&S Ref:

10.1.17 (corrected)

Permalink:

http://dlmf.nist.gov/10.49.E7

Encodings:

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See also:

Annotations for §10.49(i), §10.49 and Ch.10

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§10.49(ii) Modified Functions

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§10.49(iv) Sums or Differences of Squares

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3: 27.13 Functions

… ►Explicit formulas for $r_k\left(n\right)$ have been obtained by similar methods for $k=6,8,10$, and $12$, but they are more complicated. …Also, Milne (1996, 2002) announce new infinite families of explicit formulas extending Jacobi’s identities. …

4: 3.8 Nonlinear Equations

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§3.8(iv) Zeros of Polynomials

… ►Explicit formulas for the zeros are available if $n\le 4$; see §§1.11(iii) and 4.43. No explicit general formulas exist when $n\ge 5$. …

5: Bibliography H

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An explicit formula for Bernoulli numbers. Rep. Fac. Sci. Technol. Meijo Univ. 29, pp. 1–6.

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External Links:

ZentralBlatt

Referenced by:

§24.6.

Encodings:

BibTeX

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Explicit formulas for degenerate Bernoulli numbers. Discrete Math. 162 (1-3), pp. 175–185.

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External Links:

ISSN 0012-365X, Document, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.16(i).

Encodings:

BibTeX

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6: 18.11 Relations to Other Functions

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§18.11(i) Explicit Formulas

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7: Bibliography N

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An explicit formula for the coefficients in Laplace’s method. Constr. Approx. 38 (3), pp. 471–487.

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External Links:

ISSN 0176-4276, Document, Link, MathReview (Pedro J. Pagola), ZentralBlatt

Referenced by:

§5.11(i), §5.11(i).

Encodings:

BibTeX

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8: 29.15 Fourier Series and Chebyshev Series

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29.15.50 $${\mathrm{𝑠𝑐𝑑𝐸}}_{2n+3}^m(z,k^2)=\mathrm{cn}(z,k)\mathrm{dn}(z,k)\sum _{p=0}^n{D}_{2p+2}{U}_{2p+1}\left(\mathrm{sn}(z,k)\right).$$

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Symbols:

$U_n\left(x\right)$: Chebyshev polynomial of the second kind, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝑠𝑐𝑑𝐸}}_{2n+3}^m(z,k^2)$: Lamé polynomial, $m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter and $D_{2p+1}$: coefficents

Permalink:

http://dlmf.nist.gov/29.15.E50

Encodings:

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See also:

Annotations for §29.15(ii), §29.15 and Ch.29

►For explicit formulas for Lamé polynomials of low degree, see Arscott (1964b, p. 205).

9: 10.74 Methods of Computation

… ►In the case of the spherical Bessel functions the explicit formulas given in §§10.49(i) and 10.49(ii) are terminating cases of the asymptotic expansions given in §§10.17(i) and 10.40(i) for the Bessel functions and modified Bessel functions. …

10: Bibliography T

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Explicit formulas for the Bernoulli and Euler polynomials and numbers. Abh. Math. Sem. Univ. Hamburg 61, pp. 175–180.

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External Links:

ISSN 0025-5858, Document, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.4(v), §24.6.

Encodings:

BibTeX

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