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For the Bernoulli polynomials, the number of crossings of the x-axis in the unit interval does not go up with the degree. In the limit of large degree, they approach, when appropriately scaled, the sine and cosine functions. A similar set of polynomials, based on a generating function, is the family of Euler polynomials.
() is a Bernoulli polynomial. is a Bernoulli number, and here, = . is an Euler number. is the Riemann zeta function. ...
The Bernoulli polynomials may be defined recursively by B 0 (x) = 1 and, for k ≥ 1, ′ = (), = The periodized Bernoulli functions are defined as = (⌊ ⌋), where ⌊x⌋ denotes the largest integer less than or equal to x, so that x − ⌊x⌋ always lies in the interval [0,1).
Gregory coefficients G n, also known as reciprocal logarithmic numbers, Bernoulli numbers of the second kind, ... Jordan [1] [16] [31] defines polynomials ...
4.4 Series with Gregory's coefficients, Cauchy numbers and Bernoulli polynomials of the second kind. 5 Reflection formula. 6 Recurrence formula and characterization.
The generating function of the Bernoulli polynomials is given by: = = ()! These polynomials are given in terms of the Hurwitz zeta function: (,) = = (+)by (,) = for .Using the Ramanujan master theorem and the generating function of Bernoulli polynomials one has the following integral representation: [6]
In mathematics, the Bernoulli numbers B n are a sequence of rational numbers which occur frequently in analysis.The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of m-th powers of the first n positive integers, in the Euler–Maclaurin formula, and in expressions for certain ...
The transfer operator of the Bernoulli map = ⌊ ⌋ is exactly solvable and is a classic example of deterministic chaos; the discrete eigenvalues correspond to the Bernoulli polynomials. This operator also has a continuous spectrum consisting of the Hurwitz zeta function .