BernoulliB

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

Values at fixed points

General characteristics

Domain and analyticity

(Subscript[B, n]) is a nonanalytical function which is defined only for nonnegative integer n.

Symmetries and periodicities

Symmetry

No symmetry

Periodicity

No periodicity

Series representations

Generalized power series

Victor Adamchik

Asymptotic series expansions

Residue representations

Integral representations

On the real axis

Of the direct function

Contour integral representations

Limit representations

Continued fraction representations

Generating functions

Identities

Functional identities

G.Huvent (2006)

Identities involving determinants

Complex characteristics

Real part

Imaginary part

Absolute value

Argument

Conjugate value

Signum value

Summation

Finite summation

G.Huvent (2006)

Infinite summation

Operations

Limit operation

Representations through more general functions

Through other functions

Involving some hypergeometric-type functions

Involving polylog

Involving Stirling numbers

Involving zeta functions

Representations through equivalent functions

With related functions

Inequalities

Zeros

Other identities

Congruence properties

Divisibility properties of denominators

Theorems

The Euler-Maclaurin formula

(∫_a^(a+m h))f(x)dx≈h(((f(a))/2)+(∑_(i=1)^(m-1))f(a+i h)+((f(a+m h))/2))+(∑_(k=1)^n)(((((-1))^k)|(Subscript[B, 2k])|(h^(2k)))/((2k)!))((f^((2k-1)))(a+m h)-(f^((2k-1)))(a))/;                                      k==1,2,…,n

History

– J. Faulhaber (1631) gave the first 8 Bernoulli numbers
– Jac. Bernoulli (1705, 1713); T. Seki (1712)
– A. de Moivre (1730) found the recursion relation 
– L. Euler (1755, 1769, 1781) employed name Bernoullian numbers
– G.S.Klügel (1823) and K.G.C. von Staudt (1840) employed the notation (B^((n)))
– J. L. Raabe (1848,1851) used the name "Bernoulli polynomials"
– J. Binet (1839), Martin Ohm (1840), J.L. Raabe (1851), C. Hermite (1876),  
   M.A. Stern (1878), J.C. Adams (1878) and G. Peano (1903) introduced modern 
   notations
– C. Hermite (1875); C.J. Malmstén (1884)

Copyright