Hadamard product (of power series)
E334043
The Hadamard product (of power series) is an operation that forms a new power series by multiplying the corresponding coefficients of two given power series term by term.
All labels observed (5)
How this entity was disambiguated
This entity first appeared as the object of triple T3167254 — 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: Hadamard product (of power series) Context triple: [Jacques Hadamard, knownFor, Hadamard product (of power series)]
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A.
Cauchy–Hadamard theorem
The Cauchy–Hadamard theorem is a fundamental result in complex analysis that characterizes the radius of convergence of a power series in terms of the growth rate of its coefficients.
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B.
Lambert series
Lambert series are special infinite series in number theory and analysis, often involving arithmetic functions and powers of a variable, introduced by Johann Heinrich Lambert and used in the study of modular forms and q-series.
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C.
Hahn series
Hahn series are formal power series with exponents in an ordered abelian group and well-ordered supports, providing a general framework for constructing large ordered fields that include structures like the surreal numbers.
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D.
Pochhammer symbol
The Pochhammer symbol is a mathematical notation representing rising factorials, widely used in series expansions, special functions, and hypergeometric functions.
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E.
Kronecker product
The Kronecker product is a matrix operation that forms a large block matrix from two smaller matrices and is widely used in linear algebra, quantum computing, and signal processing.
- 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: Hadamard product (of power series) Target entity description: The Hadamard product (of power series) is an operation that forms a new power series by multiplying the corresponding coefficients of two given power series term by term.
-
A.
Cauchy–Hadamard theorem
The Cauchy–Hadamard theorem is a fundamental result in complex analysis that characterizes the radius of convergence of a power series in terms of the growth rate of its coefficients.
-
B.
Lambert series
Lambert series are special infinite series in number theory and analysis, often involving arithmetic functions and powers of a variable, introduced by Johann Heinrich Lambert and used in the study of modular forms and q-series.
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C.
Hahn series
Hahn series are formal power series with exponents in an ordered abelian group and well-ordered supports, providing a general framework for constructing large ordered fields that include structures like the surreal numbers.
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D.
Pochhammer symbol
The Pochhammer symbol is a mathematical notation representing rising factorials, widely used in series expansions, special functions, and hypergeometric functions.
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E.
Kronecker product
The Kronecker product is a matrix operation that forms a large block matrix from two smaller matrices and is widely used in linear algebra, quantum computing, and signal processing.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
binary operation on power series
ⓘ
operation on analytic functions ⓘ operation on formal power series ⓘ |
| actsOn |
exponential generating functions
ⓘ
ordinary generating functions ⓘ |
| alsoKnownAs |
Hadamard product (of power series)
ⓘ
surface form:
Hadamard multiplication (of power series)
Hadamard product (of power series) ⓘ
surface form:
Schur product (of power series)
coefficientwise product (of power series) ⓘ |
| canBeExtendedTo |
Dirichlet series via coefficientwise multiplication
ⓘ
Laurent series ⓘ |
| closureProperty | set of all power series over a field is closed under Hadamard product ⓘ |
| coefficientRule | the nth coefficient of the product equals the product of the nth coefficients of the factors ⓘ |
| definedOn | pairs of power series ⓘ |
| definition |
Hadamard product (of power series)
self-linksurface differs
ⓘ
surface form:
Given two power series f(z)=∑ a_n z^n and g(z)=∑ b_n z^n, their Hadamard product is (f*g)(z)=∑ a_n b_n z^n
|
| differsFrom | Cauchy product, which convolves coefficients instead of multiplying them termwise ⓘ |
| domain | typically defined over power series with coefficients in a field such as ℝ or ℂ ⓘ |
| field |
algebra
ⓘ
complex analysis ⓘ functional analysis ⓘ |
| formsAlgebraicStructure | commutative algebra over the base field ⓘ |
| historicalNote | introduced in the context of entire functions and their factorization by Jacques Hadamard ⓘ |
| identityElement | power series with all coefficients equal to 1 ⓘ |
| inputType |
convergent power series
ⓘ
formal power series ⓘ |
| isAssociative | true ⓘ |
| isCommutative | true ⓘ |
| isDistributiveOver | addition of power series ⓘ |
| linearity | bilinear over the base field ⓘ |
| namedAfter | Jacques Hadamard ⓘ |
| operationType | termwise multiplication of coefficients ⓘ |
| outputType | power series ⓘ |
| preserves | radius of convergence at least as small as the minimum of the radii of convergence of the factors ⓘ |
| property |
if one factor has finite support, the Hadamard product is a finite linear combination of shifts of the other series
ⓘ
if one factor is the constant series 1, the Hadamard product equals the other factor ⓘ if one factor is the zero series, the Hadamard product is the zero series ⓘ |
| relatedConcept |
Hadamard product (of power series)
self-linksurface differs
ⓘ
surface form:
Hadamard product (of matrices)
Schur product theorem ⓘ |
| relatedTo | Cauchy product of power series ⓘ |
| symbol |
*
ⓘ
∘ ⓘ ⊙ ⓘ |
| usedIn |
generating function techniques in combinatorics
ⓘ
operator theory ⓘ study of analytic continuation ⓘ study of singularities of analytic functions ⓘ |
| zeroElement | zero power series ⓘ |
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
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Input
Subject: Hadamard product (of power series) Description of subject: The Hadamard product (of power series) is an operation that forms a new power series by multiplying the corresponding coefficients of two given power series term by term.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Hadamard multiplication (of power series)
this entity surface form:
Schur product (of power series)
Hadamard product (of power series)
→
definition
→
Hadamard product (of power series)
self-linksurface differs
ⓘ
this entity surface form:
Given two power series f(z)=∑ a_n z^n and g(z)=∑ b_n z^n, their Hadamard product is (f*g)(z)=∑ a_n b_n z^n
Hadamard product (of power series)
→
relatedConcept
→
Hadamard product (of power series)
self-linksurface differs
ⓘ
this entity surface form:
Hadamard product (of matrices)