# as Bernoulli or Euler polynomials

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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. …
###### Euler Numbers and Polynomials
The notations $E_{n}$, $E_{n}\left(x\right)$, as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …
##### 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)). Euler polynomials also appear in statistical physics as well as in semi-classical approximations to quantum probability distributions (Ballentine and McRae (1998)).
##### 3: 24.3 Graphs Figure 24.3.1: Bernoulli polynomials B n ⁡ ( x ) , n = 2 , 3 , … , 6 . Magnify Figure 24.3.2: Euler polynomials E n ⁡ ( x ) , n = 2 , 3 , … , 6 . Magnify
##### 5: 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)).
##### 6: 24.13 Integrals
###### §24.13(i) BernoulliPolynomials
24.13.3 $\int_{x}^{x+(1/2)}B_{n}\left(t\right)\mathrm{d}t=\frac{E_{n}\left(2x\right)}{2% ^{n+1}},$
###### §24.13(ii) EulerPolynomials
24.13.8 $\int_{0}^{1}E_{n}\left(t\right)\mathrm{d}t=-2\frac{E_{n+1}\left(0\right)}{n+1}% =\frac{4(2^{n+2}-1)}{(n+1)(n+2)}B_{n+2},$
##### 9: 24.14 Sums
###### §24.14(i) Quadratic Recurrence Relations
24.14.4 $\sum_{k=0}^{n}{n\choose k}E_{k}E_{n-k}=-2^{n+1}E_{n+1}\left(0\right)=-2^{n+2}(% 1-2^{n+2})\frac{B_{n+2}}{n+2}.$
24.14.5 $\sum_{k=0}^{n}{n\choose k}E_{k}\left(h\right)B_{n-k}\left(x\right)=2^{n}B_{n}% \left(\tfrac{1}{2}(x+h)\right),$
For other sums involving Bernoulli and Euler numbers and polynomials see Hansen (1975, pp. 331–347) and Prudnikov et al. (1990, pp. 383–386).
##### 10: 24.17 Mathematical Applications
###### §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)).