BernoulliB2

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed n

For fixed z

General characteristics

Domain and analyticity

(Subscript[B, n])(z) is a polynomial of z and as such an analytical function of z. (Subscript[B, n])(z) is defined in the whole complex z -plane and for n∈ℕ.

Symmetries and periodicities

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to z

The function (Subscript[B, n])(z) has a pole of order n at z==(OverTilde[∞]).

Branch points

With respect to z

The function (Subscript[B, n])(z) does not have branch points.

Branch cuts

With respect to z

The function (Subscript[B, n])(z) does not have branch cuts.

Series representations

Generalized power series

Expansions at generic point z==(Subscript[z, 0])

Expansions at z==0

Exponential Fourier series

Asymptotic series expansions

Other series representations

Integral representations

On the real axis

Of the direct function

Generating functions

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Addition formulas

Multiple arguments

Identities

Recurrence identities

Consecutive neighbors

Distant neighbors

Functional identities

Relations of special kind

Complex characteristics

Real part

Imaginary part

Absolute value

Argument

Conjugate value

Signum value

Differentiation

Low-order differentiation

Symbolic differentiation

Fractional integro-differentiation

Integration

Indefinite integration

Involving only one direct function

Definite integration

For the direct function itself

Integral transforms

Fourier exp transforms

Inverse Fourier exp transforms

Fourier cos transforms

Fourier sin transforms

Laplace transforms

Summation

Finite summation

Infinite summation

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

For each n∈(SuperPlus[ℕ]) the equation (Subscript[B, n])(z)==0 has exactly n different real roots. If (n-1)/2∈(SuperPlus[ℕ]), 
the set of roots include 0, 1/2, and 1.

For each n∈ℕ the equation (Subscript[B, 2n+1])(z)==0 has in the interval (0,1) only the zero z=1/2.

Other identities

Congruence properties

Theorems

Eigenfunctions of the Fröbenius-Perron operator

Bernoulli polynomials are the right eigenfunctions of the Fröbenius-Perron operator of the r -adic map (Subscript[x, n+1])==r (Subscript[x, n])mod 1.

History

– Jac. Bernoulli (1705, 1713)
– T. Seki (1712)
– L. Euler (1738, 1755)
– .L. Raabe (1848, 1851) used the name "Bernoulli polynomials"
– C. Hermite (1875); C. J. Malmstén (1884)
– G. Peano (1903) introduced modern notations

Copyright