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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.
Download as PDF; Printable version; ... () is a Bernoulli polynomial. is a Bernoulli number, and here, =. is an Euler number. is the ...
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).
Under this convention, the polynomials form a Sheffer sequence. The Bernoulli polynomials of the second kind were largely studied by the Hungarian mathematician Charles Jordan, [1] [2] but their history may also be traced back to the much earlier works. [3]
and call them Bernoulli polynomials of the second kind. From the above, it is clear that G n = ψ n (0) . Carlitz [ 16 ] generalized Jordan's polynomials ψ n ( s ) by introducing polynomials β
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]
Faulhaber's formula is also called Bernoulli's formula. Faulhaber did not know the properties of the coefficients later discovered by Bernoulli. Rather, he knew at least the first 17 cases, as well as the existence of the Faulhaber polynomials for odd powers described below. [2] Jakob Bernoulli's Summae Potestatum, Ars Conjectandi, 1713
Here the are the Bernoulli numbers and is the Glaisher–Kinkelin constant. (Note that somewhat confusingly at the time of Barnes [ 7 ] the Bernoulli number B 2 k {\displaystyle B_{2k}} would have been written as ( − 1 ) k + 1 B k {\displaystyle (-1)^{k+1}B_{k}} , but this convention is no longer current.)