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Bernoulli and Euler polynomials

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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
Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. The notations E n , E n ( x ) , as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …
2: 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
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: 24.4 Basic Properties
§24.4(i) Difference Equations
§24.4(ii) Symmetry
§24.4(iii) Sums of Powers
Let P ( x ) denote any polynomial in x , and after expanding set ( B ( x ) ) n = B n ( x ) and ( E ( x ) ) n = E n ( x ) . …
24.4.39 E n ( x + h ) = ( E ( x ) + h ) n .
5: 24.21 Software
§24.21(ii) B n , B n ( x ) , E n , and E n ( x )
6: 24.13 Integrals
24.13.3 x x + ( 1 / 2 ) B n ( t ) d t = E n ( 2 x ) 2 n + 1 ,
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 ,
24.13.9 0 1 / 2 E 2 n ( t ) d t = E 2 n + 1 ( 0 ) 2 n + 1 = 2 ( 2 2 n + 2 1 ) B 2 n + 2 ( 2 n + 1 ) ( 2 n + 2 ) ,
24.13.11 0 1 E n ( t ) E m ( t ) d t = ( 1 ) n 4 ( 2 m + n + 2 1 ) m ! n ! ( m + n + 2 ) ! B m + n + 2 .
§24.13(iii) Compendia
7: 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 : …
24.16.13 E n ( x ) = 2 1 n n + 1 B n + 1 , χ 4 ( 2 x 1 ) .
§24.16(iii) Other Generalizations
8: 24.2 Definitions and Generating Functions
Table 24.2.2: Bernoulli and Euler polynomials.
n B n ( x ) E n ( x )
9: 24.17 Mathematical Applications
§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)).
10: 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).