Search results
Results from the WOW.Com Content Network
A periodic Bernoulli polynomial P n (x) is a Bernoulli polynomial evaluated at the fractional part of the argument x. These functions are used to provide the remainder term in the Euler–Maclaurin formula relating sums to integrals. The first polynomial is a sawtooth function.
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 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).
In 1713, Jacob Bernoulli published under the title Summae Potestatum an expression of the sum of the p powers of the n first integers as a (p + 1)th-degree polynomial function of n, with coefficients involving numbers B j, now called Bernoulli numbers:
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 ...
() is a Bernoulli polynomial. is a Bernoulli number, and here, =. is an Euler ... is the Touchard polynomials. Trigonometric, inverse trigonometric, hyperbolic, and ...
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
Series with Gregory's coefficients, Cauchy numbers and Bernoulli polynomials of the second kind [ edit ] There exist various series for the digamma containing rational coefficients only for the rational arguments.