In mathematics, the Bernoulli polynomials, named after Jacob Bernoulli, combine the Bernoulli numbers and binomial coefficients. They are used for series expansion of functions, and with the Euler–MacLaurin formula.
These polynomials occur in the study of many special functions and, in particular, the Riemann zeta function and the Hurwitz zeta function. They are an Appell sequence (i.e. a Sheffer sequence for the ordinary derivative operator). 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.
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functions. A similar set of polynomials, based on a generating function, is the family of Euler polynomials. The Bernoullipolynomials Bn can be defined by a...
divisible by 4 and positive otherwise. The Bernoulli numbers are special values of the Bernoullipolynomials B n ( x ) {\displaystyle B_{n}(x)} , with...
The Bernoullipolynomials of the second kind ψn(x), also known as the Fontana-Bessel polynomials, are the polynomials defined by the following generating...
Bernoulli family of Basel. Bernoulli differential equation Bernoulli distribution Bernoulli number BernoullipolynomialsBernoulli process Bernoulli Society...
In probability and statistics, a Bernoulli process (named after Jacob Bernoulli) is a finite or infinite sequence of binary random variables, so it is...
The Bernoullipolynomials may be obtained as a special case of the Hurwitz zeta function, and thus the identities follow from there. The Bernoulli map...
SL-type Clausen function are polynomials in θ {\displaystyle \,\theta \,} , and are closely related to the Bernoullipolynomials. This connection is apparent...
Stirling numbers, the Bernoulli numbers, and the generalized Bernoullipolynomials. There are multiple variants of the Stirling polynomial sequence considered...
2024. Bernoulli differential equation Bernoulli distribution Bernoulli number BernoullipolynomialsBernoulli process Bernoulli trial Bernoulli's principle...
}{\frac {C_{n}(n-1)!}{(v)_{n}}},\qquad \Re (v)>1,} A series with the Bernoullipolynomials of the second kind has the following form ψ ( v ) = ln ( v + a...
Jacob Bernoulli (also known as James in English or Jacques in French; 6 January 1655 [O.S. 27 December 1654] – 16 August 1705) was one of the many prominent...
Bernoulli number BernoullipolynomialsBernoulli process Bernoulli trial Lemniscate of BernoulliBernoulli, a journal published by the Bernoulli Society for...
where the B n {\displaystyle B_{n}} are the Bernoullipolynomials. This follows because the Bernoullipolynomials obey the identity 1 2 B n ( y 2 ) + 1 2...
taken literally without logical difficulty. An example involves the Bernoullipolynomials. Consider, for example, the ordinary binomial expansion (which contains...
Abel polynomials; The Bernoullipolynomials; The Euler polynomial; The central factorial polynomials; The Hermite polynomials; The Laguerre polynomials; The...
{\displaystyle \{x^{n}\}} are the Hermite polynomials, the Bernoullipolynomials, and the Euler polynomials. Every Appell sequence is a Sheffer sequence...
deterministic chaos; the discrete eigenvalues correspond to the Bernoullipolynomials. This operator also has a continuous spectrum consisting of the...
x} B n ( x ) {\displaystyle B_{n}(x)} is a Bernoullipolynomial. B n {\displaystyle B_{n}} is a Bernoulli number, and here, B 1 = − 1 2 . {\displaystyle...
Bernstein polynomials, restricted to the interval [0, 1], became important in the form of Bézier curves. A numerically stable way to evaluate polynomials in...
ISBN 978-2-88124-682-1. (see § 1.2, "The generalized zeta function, Bernoullipolynomials, Euler polynomials, and polylogarithms", p. 23.) Robinson, J.E. (1951). "Note...
Since Bernoullipolynomials is a generalization of Bernoulli numbers, exponentiation of Bernoulli umbra can be expressed via Bernoullipolynomials: eval...
call them Bernoullipolynomials of the second kind. From the above, it is clear that Gn = ψn(0). Carlitz generalized Jordan's polynomials ψn(s) by introducing...