Bernoulli polynomials
E583311
Bernoulli polynomials are a sequence of polynomials deeply connected to number theory and analysis, appearing in the study of special functions, series expansions, and the evaluation of sums of powers of integers.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf | sequence of polynomials ⓘ |
| AppellProperty | derivative lowers index: B_n'(x) = n B_{n-1}(x) ⓘ |
| application |
Euler–Maclaurin summation formula
NERFINISHED
ⓘ
Faulhaber formulas for sums of powers of integers NERFINISHED ⓘ Fourier series expansions ⓘ approximation theory ⓘ asymptotic expansions of sums and integrals ⓘ evaluation of Σ_{k=1}^n k^p ⓘ special values of the Riemann zeta function ⓘ |
| B0 | B_0(x) = 1 ⓘ |
| B1 | B_1(x) = x - 1/2 ⓘ |
| B2 | B_2(x) = x^2 - x + 1/6 ⓘ |
| B3 | B_3(x) = x^3 - 3/2 x^2 + 1/2 x ⓘ |
| B4 | B_4(x) = x^4 - 2 x^3 + x^2 - 1/30 ⓘ |
| basisProperty | form a basis of the vector space of polynomials ⓘ |
| degree | deg B_n(x) = n ⓘ |
| differentialEquation | B_n^{(k)}(x) = n(n-1)…(n-k+1) B_{n-k}(x) ⓘ |
| domain | complex variable x ⓘ |
| expansionProperty | any polynomial can be expressed as a linear combination of Bernoulli polynomials ⓘ |
| field | mathematics ⓘ |
| functionalEquation | ∫_0^1 B_n(x) dx = 0 for n ≥ 1 ⓘ |
| generalizationOf | Bernoulli numbers NERFINISHED ⓘ |
| generatingFunction | t e^{xt} / (e^t - 1) = Σ_{n=0}^∞ B_n(x) t^n / n! ⓘ |
| leadingCoefficient | 1 ⓘ |
| namedAfter | Jacob Bernoulli NERFINISHED ⓘ |
| notation | B_n(x) ⓘ |
| orthogonalityProperty | not orthogonal on standard intervals with usual weights ⓘ |
| parameter | n ⓘ |
| parameterType | nonnegative integer ⓘ |
| parityProperty | B_{2m+1}(0) = 0 for m ≥ 1 ⓘ |
| recurrenceRelation |
B_0(x) = 1
ⓘ
B_n'(x) = n B_{n-1}(x) ⓘ B_n(x) = Σ_{k=0}^n {n \\ k} B_k x^{n-k} with suitable coefficients ⓘ B_n(x+1) - B_n(x) = n x^{n-1} ⓘ |
| relatedConcept |
Appell sequence
ⓘ
Bernoulli numbers NERFINISHED ⓘ Hurwitz zeta function NERFINISHED ⓘ |
| relationToBernoulliNumbers | B_n = B_n(0) ⓘ |
| specialValue |
B_1(0) = -1/2
ⓘ
B_1(1) = 1/2 ⓘ B_n(0) = B_n (Bernoulli number) ⓘ B_n(1) = B_n for n ≠ 1 ⓘ |
| subfield |
mathematical analysis
ⓘ
number theory ⓘ |
| symmetryProperty | B_n(1-x) = (-1)^n B_n(x) ⓘ |
| usedIn |
analytic number theory
ⓘ
theory of special functions ⓘ |
| zetaRelation | ζ(-n,x) = - B_{n+1}(x)/(n+1) ⓘ |
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.