Bernoulli

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1: 24.1 Special Notation
Bernoulli Numbers and Polynomials
The origin of the notation $B_{n}$, $B_{n}\left(x\right)$, is not clear. …
2: 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)). …
3: 24.3 Graphs Figure 24.3.1: Bernoulli polynomials B n ⁡ ( x ) , n = 2 , 3 , … , 6 . Magnify
6: 24.19 Methods of Computation
§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 $\pmod{p}$ for Bernoulli numbers in terms of sums of powers. See also §24.10(iii).

• 7: 24.14 Sums
§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).
8: 24.13 Integrals
§24.13(i) Bernoulli Polynomials
24.13.4 $\int_{0}^{1/2}B_{n}\left(t\right)\mathrm{d}t=\frac{1-2^{n+1}}{2^{n}}\frac{B_{n% +1}}{n+1},$
24.13.6 $\int_{0}^{1}B_{n}\left(t\right)B_{m}\left(t\right)\mathrm{d}t=\frac{(-1)^{n-1}% m!n!}{(m+n)!}B_{m+n}.$
For integrals of the form $\int_{0}^{x}B_{n}\left(t\right)B_{m}\left(t\right)\mathrm{d}t$ and $\int_{0}^{x}B_{n}\left(t\right)B_{m}\left(t\right)B_{k}\left(t\right)\mathrm{d}t$ see Agoh and Dilcher (2011). …
10: 24.5 Recurrence Relations
§24.5 Recurrence Relations
24.5.1 $\sum_{k=0}^{n-1}{n\choose k}B_{k}\left(x\right)=nx^{n-1},$ $n=2,3,\dots$,
24.5.3 $\sum_{k=0}^{n-1}{n\choose k}B_{k}=0,$ $n=2,3,\dots$,