Bernoulli monosplines

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1: 24.1 Special Notation

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Bernoulli Numbers and Polynomials

►The origin of the notation $B_n$, $B_n\left(x\right)$, is not clear. …

2: 24.17 Mathematical Applications

… ►The functions … ►

Bernoulli Monosplines

… ► $M_n(x)$ is a monospline of degree $n$, and it follows from (24.4.25) and (24.4.27) that …For each $n=1,2,\mathrm{\dots }$ the function $M_n(x)$ is also the unique cardinal monospline of degree $n$ satisfying (24.17.6), provided that … ► …

3: 24.18 Physical Applications

§24.18 Physical Applications

►Bernoulli polynomials appear in statistical physics (Ordóñez and Driebe (1996)), in discussions of Casimir forces (Li et al. (1991)), and in a study of quark-gluon plasma (Meisinger et al. (2002)). …

4: 24.3 Graphs

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Figure 24.3.1: Bernoulli polynomials $B_n\left(x\right)$, $n=2,3,\mathrm{\dots },6$. Magnify

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5: 24.4 Basic Properties

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§24.4(ii) Symmetry

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§24.4(v) Multiplication Formulas

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Raabe’s Theorem

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§24.4(vii) Derivatives

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§24.4(ix) Relations to Other Functions

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6: 24.16 Generalizations

§24.16 Generalizations

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Bernoulli Numbers of the Second Kind

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Degenerate Bernoulli Numbers

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§24.16(ii) Character Analogs

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§24.16(iii) Other Generalizations

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7: 24.19 Methods of Computation

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§24.19(i) Bernoulli and Euler Numbers and Polynomials

… ►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). ►

§24.19(ii) Values of $B_n$ Modulo $p$

… ►We list here three methods, arranged in increasing order of efficiency. ►

Tanner and Wagstaff (1987) derives a congruence $(modp)$ for Bernoulli numbers in terms of sums of powers. See also §24.10(iii).

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8: 24.14 Sums

§24.14 Sums

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§24.14(i) Quadratic Recurrence Relations

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24.14.2 $$\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)B_k{B}_{n-k}=(1-n)B_n-n{B}_{n-1}.$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/24.14.E2

Encodings:

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See also:

Annotations for §24.14(i), §24.14 and Ch.24

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§24.14(ii) Higher-Order Recurrence Relations

… ►For other sums involving Bernoulli and Euler numbers and polynomials see Hansen (1975, pp. 331–347) and Prudnikov et al. (1990, pp. 383–386).

9: 24.13 Integrals

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§24.13(i) Bernoulli Polynomials

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24.13.4 $${\int }_0^{1/2}{B}_n\left(t\right)dt=\frac{1-2^{n+1}}{2^n}\frac{B_{n+1}}{n+1},$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/24.13.E4

Encodings:

pMML, png, TeX

See also:

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

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24.13.6 $${\int }_0^1{B}_n\left(t\right)B_m\left(t\right)dt=\frac{{(-1)}^{n-1}m!n!}{(m+n)!}{B}_{m+n}.$$

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Symbols:

$B_n$: Bernoulli numbers, $B_n\left(x\right)$: Bernoulli 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(i)

Permalink:

http://dlmf.nist.gov/24.13.E6

Encodings:

pMML, png, TeX

See also:

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

►For integrals of the form ${\int }_0^x{B}_n\left(t\right)B_m\left(t\right)dt$ and ${\int }_0^x{B}_n\left(t\right)B_m\left(t\right)B_k\left(t\right)dt$ see Agoh and Dilcher (2011). … ►

§24.13(iii) Compendia

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10: 24.2 Definitions and Generating Functions

§24.2 Definitions and Generating Functions

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§24.2(i) Bernoulli Numbers and Polynomials

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§24.2(iii) Periodic Bernoulli and Euler Functions

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Table 24.2.1: Bernoulli and Euler numbers.

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$n$	$B_n$	$E_n$
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