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Bernoulli monosplines

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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: 24.17 Mathematical Applications
The functions …
Bernoulli Monosplines
M n ( x ) is a monospline of degree n , and it follows from (24.4.25) and (24.4.27) that …For each n = 1 , 2 , the function M n ( x ) is also the unique cardinal monospline of degree n satisfying (24.17.6), provided that …
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: 24.3 Graphs
See accompanying text
Figure 24.3.1: Bernoulli polynomials B n ( x ) , n = 2 , 3 , , 6 . Magnify
5: 24.4 Basic Properties
§24.4(ii) Symmetry
§24.4(v) Multiplication Formulas
Raabe’s Theorem
§24.4(vii) Derivatives
§24.4(ix) Relations to Other Functions
6: 24.16 Generalizations
§24.16 Generalizations
Bernoulli Numbers of the Second Kind
Degenerate Bernoulli Numbers
§24.16(ii) Character Analogs
§24.16(iii) Other Generalizations
7: 24.19 Methods of Computation
§24.19(i) Bernoulli and Euler Numbers and Polynomials
For algorithms for computing B n , E n , B n ( x ) , and E n ( x ) 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 ( mod p ) for Bernoulli numbers in terms of sums of powers. See also §24.10(iii).

  • 8: 24.14 Sums
    §24.14 Sums
    §24.14(i) Quadratic Recurrence Relations
    §24.14(ii) Higher-Order Recurrence Relations
    24.14.8 ( 2 n ) ! ( 2 j ) ! ( 2 k ) ! ( 2 ) ! B 2 j B 2 k B 2 = ( n - 1 ) ( 2 n - 1 ) B 2 n + n ( n - 1 2 ) B 2 n - 2 ,
    For other sums involving Bernoulli and Euler numbers and polynomials see Hansen (1975, pp. 331–347) and Prudnikov et al. (1990, pp. 383–386).
    9: 24.13 Integrals
    §24.13(i) Bernoulli Polynomials
    24.13.4 0 1 / 2 B n ( t ) d t = 1 - 2 n + 1 2 n B n + 1 n + 1 ,
    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
    10: 24.2 Definitions and Generating Functions
    §24.2 Definitions and Generating Functions
    §24.2(i) Bernoulli Numbers and Polynomials
    §24.2(iii) Periodic Bernoulli and Euler Functions
    Table 24.2.1: Bernoulli and Euler numbers.
    n B n E n