Bernoulli Polynomial
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A Bernoulli Polynomial is a orthogonal polynomial that is an Appell sequence.
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- Example(s):
- See: Euler Number, Special Functions, Riemann Zeta Function, Hurwitz Zeta Function, Sheffer Sequence, Derivative, Trigonometric Function, Bernoulli Number, Taylor Polynomial.
References
2016
- (Wikipedia, 2016) ⇒ https://en.wikipedia.org/wiki/Bernoulli_polynomials Retrieved:2016-10-23.
- In mathematics, the Bernoulli polynomials occur in the study of many special functions and in particular the Riemann zeta function and the Hurwitz zeta function. This is in large part because they are an Appell sequence, i.e. a Sheffer sequence for the ordinary derivative operator. Unlike orthogonal polynomials, the Bernoulli polynomials are remarkable in that the number of crossings of the x-axis in the unit interval does not go up as the degree of the polynomials goes up. In the limit of large degree, the Bernoulli polynomials, appropriately scaled, approach the sine and cosine functions.
This article also discusses the Bernoulli polynomials and the related Euler polynomials, and the Bernoulli and Euler numbers.
- In mathematics, the Bernoulli polynomials occur in the study of many special functions and in particular the Riemann zeta function and the Hurwitz zeta function. This is in large part because they are an Appell sequence, i.e. a Sheffer sequence for the ordinary derivative operator. Unlike orthogonal polynomials, the Bernoulli polynomials are remarkable in that the number of crossings of the x-axis in the unit interval does not go up as the degree of the polynomials goes up. In the limit of large degree, the Bernoulli polynomials, appropriately scaled, approach the sine and cosine functions.