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1: 24.1 Special Notation
Bernoulli Numbers and Polynomials
The origin of the notation B n , B n ( x ) , is not clear. …
Euler Numbers and Polynomials
The notations E n , E n ( x ) , 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
See accompanying text
Figure 24.3.1: Bernoulli polynomials B n ( x ) , n = 2 , 3 , , 6 . Magnify
See accompanying text
Figure 24.3.2: Euler polynomials E n ( x ) , n = 2 , 3 , , 6 . Magnify
4: 24.4 Basic Properties
§24.4(i) Difference Equations
§24.4(ii) Symmetry
§24.4(iii) Sums of Powers
§24.4(iv) Finite Expansions
§24.4(vi) Special Values
5: 24.16 Generalizations
§24.16 Generalizations
For = 0 , 1 , 2 , , Bernoulli and Euler polynomials of order are defined respectively by …When x = 0 they reduce to the Bernoulli and Euler numbers of order : …
§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) Bernoulli Polynomials
24.13.3 x x + ( 1 / 2 ) B n ( t ) d t = E n ( 2 x ) 2 n + 1 ,
§24.13(ii) Euler Polynomials
24.13.8 0 1 E n ( t ) d t = - 2 E n + 1 ( 0 ) n + 1 = 4 ( 2 n + 2 - 1 ) ( n + 1 ) ( n + 2 ) B n + 2 ,
§24.13(iii) Compendia
7: 24.21 Software
§24.21(ii) B n , B n ( x ) , E n , and E n ( x )
8: 24.2 Definitions and Generating Functions
§24.2 Definitions and Generating Functions
§24.2(i) Bernoulli Numbers and Polynomials
Table 24.2.2: Bernoulli and Euler polynomials.
n B n ( x ) E n ( x )
9: 24.14 Sums
§24.14 Sums
§24.14(i) Quadratic Recurrence Relations
24.14.4 k = 0 n ( n k ) E k E n - k = - 2 n + 1 E n + 1 ( 0 ) = - 2 n + 2 ( 1 - 2 n + 2 ) B n + 2 n + 2 .
24.14.5 k = 0 n ( n k ) E k ( h ) B n - k ( x ) = 2 n B n ( 1 2 ( x + h ) ) ,
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 Mathematical Applications
Bernoulli Monosplines
§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)).