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Bernoulli 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. …
2: 24.3 Graphs
See accompanying text
Figure 24.3.1: Bernoulli polynomials B n ( x ) , n = 2 , 3 , , 6 . Magnify
3: 24.16 Generalizations
For = 0 , 1 , 2 , , Bernoulli and Euler polynomials of order are defined respectively by …
B n ( ) = B n ( ) ( 0 ) ,
For extensions of B n ( ) ( x ) to complex values of x , n , and , and also for uniform asymptotic expansions for large x and large n , see Temme (1995b) and López and Temme (1999b, 2010b). …
24.16.6 n ! b n = - 1 n - 1 B n ( n - 1 ) , n = 2 , 3 , .
B n ( x ) is a polynomial in x of degree n . …
4: 24.4 Basic Properties
24.4.1 B n ( x + 1 ) - B n ( x ) = n x n - 1 ,
24.4.3 B n ( 1 - x ) = ( - 1 ) n B n ( x ) ,
24.4.5 ( - 1 ) n B n ( - x ) = B n ( x ) + n x n - 1 ,
24.4.21 B n ( x ) = 2 n - 1 ( B n ( 1 2 x ) + B n ( 1 2 x + 1 2 ) ) ,
24.4.37 B n ( x + h ) = ( B ( x ) + h ) n ,
5: 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)). …
6: 25.1 Special Notation
k , m , n

nonnegative integers.

B n , B n ( x )

Bernoulli number and polynomial24.2(i)).

B ~ n ( x )

periodic Bernoulli function B n ( x - x ) .

7: 24.13 Integrals
§24.13(i) Bernoulli Polynomials
24.13.1 B n ( t ) d t = B n + 1 ( t ) n + 1 + const. ,
24.13.6 0 1 B n ( t ) B m ( t ) d t = ( - 1 ) n - 1 m ! n ! ( m + n ) ! B m + n .
For integrals of the form 0 x B n ( t ) B m ( t ) d t and 0 x B n ( t ) B m ( t ) B k ( t ) d t see Agoh and Dilcher (2011). …
§24.13(iii) Compendia
8: 24.21 Software
§24.21(ii) B n , B n ( x ) , E n , and E n ( x )
9: 24.2 Definitions and Generating Functions
24.2.4 B n = B n ( 0 ) ,
B ~ n ( x ) = B n ( x ) ,
B ~ n ( x + 1 ) = B ~ n ( x ) ,
Table 24.2.2: Bernoulli and Euler polynomials.
n B n ( x ) E n ( x )
10: 24.17 Mathematical Applications
§24.17 Mathematical Applications
24.17.5 M n ( x ) = { B ~ n ( x ) - B n , n  even , B ~ n ( x + 1 2 ) , n  odd .
24.17.8 F ( x ) = B ~ n ( x ) - 2 - n B n
§24.17(iii) Number Theory