functions. A similar set of polynomials, based on a generating function, is the family of Euler polynomials. The Bernoulli polynomials Bn can be defined by a...
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divisible by 4 and positive otherwise. The Bernoulli numbers are special values of the Bernoulli polynomials B n ( x ) {\displaystyle B_{n}(x)} , with...
92 KB (12,860 words) - 23:44, 15 August 2024
The Bernoulli polynomials of the second kind ψn(x), also known as the Fontana–Bessel polynomials, are the polynomials defined by the following generating...
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Bernoulli family of Basel. Bernoulli differential equation Bernoulli distribution Bernoulli number Bernoulli polynomials Bernoulli process Bernoulli Society...
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In probability and statistics, a Bernoulli process (named after Jacob Bernoulli) is a finite or infinite sequence of binary random variables, so it is...
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Faulhaber's formula (redirect from Bernoulli's formula)
the polynomials in a on the right-hand sides of these identities Faulhaber polynomials. These polynomials are divisible by a2 because the Bernoulli number...
33 KB (7,891 words) - 23:57, 25 July 2024
Multiplication theorem (section Bernoulli polynomials)
The Bernoulli polynomials may be obtained as a special case of the Hurwitz zeta function, and thus the identities follow from there. The Bernoulli map...
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well-known Mellin inversion theorem. The generating function of the Bernoulli polynomials B k ( x ) {\textstyle B_{k}(x)} is given by: z e x z e z − 1 = ∑...
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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...
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has an exact expression in terms of the periodized Bernoulli functions Pk(x). The Bernoulli polynomials may be defined recursively by B0(x) = 1 and, for...
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Digamma function (section Series with Gregory's coefficients, Cauchy numbers and Bernoulli polynomials of the second kind)
}{\frac {C_{n}(n-1)!}{(v)_{n}}},\qquad \Re (v)>1,} A series with the Bernoulli polynomials of the second kind has the following form ψ ( v ) = ln ( v + a...
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Bernoulli number Bernoulli polynomials Bernoulli process Bernoulli trial Lemniscate of Bernoulli Bernoulli, a journal published by the Bernoulli Society for...
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2024. Bernoulli differential equation Bernoulli distribution Bernoulli number Bernoulli polynomials Bernoulli process Bernoulli trial Bernoulli's principle...
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All one polynomials Appell sequence Askey–Wilson polynomials Bell polynomials Bernoulli polynomials Bernstein polynomial Bessel polynomials Binomial...
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SL-type Clausen function are polynomials in θ {\displaystyle \,\theta \,} , and are closely related to the Bernoulli polynomials. This connection is apparent...
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All-one polynomials Abel polynomials Bell polynomials Bernoulli polynomials Cyclotomic polynomials Dickson polynomials Fibonacci polynomials Lagrange...
2 KB (176 words) - 15:36, 14 August 2021
Stirling numbers, the Bernoulli numbers, and the generalized Bernoulli polynomials. There are multiple variants of the Stirling polynomial sequence considered...
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Dyadic transformation (redirect from Bernoulli map)
where the B n {\displaystyle B_{n}} are the Bernoulli polynomials. This follows because the Bernoulli polynomials obey the identity 1 2 B n ( y 2 ) + 1 2...
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Jakob Bernoulli's honour: Bernoulli's formula Bernoulli differential equation Bernoulli's inequality Bernoulli numbers Bernoulli polynomials Bernoulli's quadrisection...
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Sheffer sequence (redirect from Sheffer polynomials)
Abel polynomials; The Bernoulli polynomials; The Euler polynomial; The central factorial polynomials; The Hermite polynomials; The Laguerre polynomials; The...
7 KB (1,049 words) - 22:05, 9 April 2024
Appell sequence (redirect from Appell polynomials)
{\displaystyle \{x^{n}\}} are the Hermite polynomials, the Bernoulli polynomials, and the Euler polynomials. Every Appell sequence is a Sheffer sequence...
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Umbral calculus (category Polynomials)
taken literally without logical difficulty. An example involves the Bernoulli polynomials. Consider, for example, the ordinary binomial expansion (which contains...
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Transfer operator (redirect from Bernoulli operator)
deterministic chaos; the discrete eigenvalues correspond to the Bernoulli polynomials. This operator also has a continuous spectrum consisting of the...
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x} B n ( x ) {\displaystyle B_{n}(x)} is a Bernoulli polynomial. B n {\displaystyle B_{n}} is a Bernoulli number, and here, B 1 = − 1 2 . {\displaystyle...
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Euler–Bernoulli beam theory (also known as engineer's beam theory or classical beam theory) is a simplification of the linear theory of elasticity which...
46 KB (7,230 words) - 13:23, 12 April 2024
Bernstein polynomials, restricted to the interval [0, 1], became important in the form of Bézier curves. A numerically stable way to evaluate polynomials in...
21 KB (3,797 words) - 00:15, 3 January 2024
Since Bernoulli polynomials is a generalization of Bernoulli numbers, exponentiation of Bernoulli umbra can be expressed via Bernoulli polynomials: eval...
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ISBN 978-2-88124-682-1. (see § 1.2, "The generalized zeta function, Bernoulli polynomials, Euler polynomials, and polylogarithms", p. 23.) Robinson, J.E. (1951). "Note...
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can be expressed in terms of special functions such as Bernoulli polynomials or Hermite polynomials. The Wick product is named after physicist Gian-Carlo...
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Barnes: Barnes G-function E. T. Bell: Bell polynomials Bender–Dunne polynomial Jacob Bernoulli: Bernoulli polynomial Friedrich Bessel: Bessel function, Bessel–Clifford...
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