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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Bernoulli polynomials canonical | 4 |
How this entity was disambiguated
This entity first appeared as the object of triple T6293470 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Bernoulli polynomials Context triple: [Bernoulli family, knownFor, Bernoulli polynomials]
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A.
Bernoulli numbers
Bernoulli numbers are a sequence of rational numbers that play a central role in number theory and analysis, especially in formulas for sums of powers of integers and in the study of special functions like the Riemann zeta function.
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B.
Bernstein polynomials
Bernstein polynomials are a family of polynomials used in approximation theory that provide a constructive proof of the Weierstrass approximation theorem by uniformly approximating continuous functions on a closed interval.
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C.
Jacobi polynomials
Jacobi polynomials are a family of classical orthogonal polynomials depending on two parameters, widely used in approximation theory, numerical analysis, and solutions of differential equations.
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D.
Orthogonal Polynomials
Orthogonal Polynomials is a classic mathematical monograph by Gábor Szegő that systematically develops the theory and applications of orthogonal polynomial systems in analysis and approximation theory.
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E.
Chebyshev functions
Chebyshev functions are arithmetic functions in number theory that encode information about the distribution of prime numbers and play a key role in analytic approaches to the prime number theorem.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Bernoulli polynomials Target entity description: 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.
-
A.
Bernoulli numbers
Bernoulli numbers are a sequence of rational numbers that play a central role in number theory and analysis, especially in formulas for sums of powers of integers and in the study of special functions like the Riemann zeta function.
-
B.
Bernstein polynomials
Bernstein polynomials are a family of polynomials used in approximation theory that provide a constructive proof of the Weierstrass approximation theorem by uniformly approximating continuous functions on a closed interval.
-
C.
Jacobi polynomials
Jacobi polynomials are a family of classical orthogonal polynomials depending on two parameters, widely used in approximation theory, numerical analysis, and solutions of differential equations.
-
D.
Orthogonal Polynomials
Orthogonal Polynomials is a classic mathematical monograph by Gábor Szegő that systematically develops the theory and applications of orthogonal polynomial systems in analysis and approximation theory.
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E.
Chebyshev functions
Chebyshev functions are arithmetic functions in number theory that encode information about the distribution of prime numbers and play a key role in analytic approaches to the prime number theorem.
- F. None of above. chosen
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
ⓘ
Faulhaber formulas for sums of powers of integers ⓘ
linked to:
Bernoulli numbers
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 ⓘ |
| generatingFunction | t e^{xt} / (e^t - 1) = Σ_{n=0}^∞ B_n(x) t^n / n! ⓘ |
| leadingCoefficient | 1 ⓘ |
| namedAfter |
Jacob Bernoulli
ⓘ
linked to:
Jakob Bernoulli
|
| 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 ⓘ Hurwitz zeta function ⓘ |
| 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) ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Bernoulli polynomials Description of subject: 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.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.