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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.
The evaluation of incomplete exponential Bell polynomial B n,k (x 1,x 2,...) on the sequence of ones equals a Stirling number of the second kind: {} =, (,, …,). Another explicit formula given in the NIST Handbook of Mathematical Functions is
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).
() is a Bernoulli polynomial. is a Bernoulli number, and here, = . is an Euler number. is the Riemann zeta function. ...
Gregory coefficients G n, also known as reciprocal logarithmic numbers, Bernoulli numbers of the second kind, ... Jordan [1] [16] [31] defines polynomials ...
Note that the polynomial in parentheses is the derivative of the polynomial above with respect to a. Since a = n(n + 1)/2, these formulae show that for an odd power (greater than 1), the sum is a polynomial in n having factors n 2 and (n + 1) 2, while for an even power the polynomial has factors n, n + 1/2 and n + 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]
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]