Editing Bernoulli polynomials (section)

{{Use American English|date = March 2019}}
{{Short description|Polynomial sequence}}

[[File:Bernoulli polynomials.svg|thumb|right|Bernoulli polynomials]]

In [[mathematics]], the '''Bernoulli polynomials''', named after [[Jacob Bernoulli]], combine the [[Bernoulli number]]s and [[binomial coefficient]]s. They are used for [[series expansion]] of [[function (mathematics)|functions]], and with the [[Euler–MacLaurin formula]].

These [[polynomial]]s 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 of a polynomial|degree]]. In the limit of large degree, they approach, when appropriately scaled, the [[trigonometric function|sine and cosine functions]].

A similar set of polynomials, based on a generating function, is the family of '''Euler polynomials'''.