Pochhammer symbol
E146427
The Pochhammer symbol is a mathematical notation representing rising factorials, widely used in series expansions, special functions, and hypergeometric functions.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Pochhammer symbol canonical | 3 |
| Pochhammer | 1 |
| q-Pochhammer symbol | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1286006 — 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: Pochhammer symbol Context triple: [generalized binomial theorem, relatedConcept, Pochhammer symbol]
-
A.
binomial theorem
The binomial theorem is a fundamental algebraic formula that provides a systematic way to expand powers of binomial expressions, playing a key role in combinatorics and mathematical analysis.
-
B.
Pascal's triangle
Pascal's triangle is a triangular array of numbers in which each entry is the sum of the two directly above it, widely used in combinatorics, algebra, and probability.
-
C.
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.
-
D.
generalized binomial theorem
The generalized binomial theorem extends the classical binomial theorem by allowing real or complex exponents, expressing powers of a binomial as an infinite series using generalized binomial coefficients.
-
E.
multinomial theorem
The multinomial theorem is a fundamental algebraic formula that generalizes the binomial theorem to express powers of sums with any number of terms using multinomial coefficients.
- 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: Pochhammer symbol Target entity description: The Pochhammer symbol is a mathematical notation representing rising factorials, widely used in series expansions, special functions, and hypergeometric functions.
-
A.
binomial theorem
The binomial theorem is a fundamental algebraic formula that provides a systematic way to expand powers of binomial expressions, playing a key role in combinatorics and mathematical analysis.
-
B.
Pascal's triangle
Pascal's triangle is a triangular array of numbers in which each entry is the sum of the two directly above it, widely used in combinatorics, algebra, and probability.
-
C.
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.
-
D.
generalized binomial theorem
The generalized binomial theorem extends the classical binomial theorem by allowing real or complex exponents, expressing powers of a binomial as an infinite series using generalized binomial coefficients.
-
E.
multinomial theorem
The multinomial theorem is a fundamental algebraic formula that generalizes the binomial theorem to express powers of sums with any number of terms using multinomial coefficients.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical notation
ⓘ
symbol ⓘ |
| alsoKnownAs |
rising factorial
ⓘ
shifted factorial ⓘ |
| appearsIn |
Taylor series of hypergeometric functions
ⓘ
binomial series expansion ⓘ solutions of differential equations ⓘ |
| belongsTo | class of shifted factorials ⓘ |
| category |
combinatorics
ⓘ
mathematical analysis ⓘ special functions ⓘ |
| definedFor |
complex parameter a
ⓘ
nonnegative integer n ⓘ |
| definition |
(a)_0 = 1
ⓘ
(a)_n = a(a+1)(a+2)\cdots(a+n-1) for n > 0 ⓘ |
| denotedBy | (a)_n ⓘ |
| distinguishedFrom | falling factorial ⓘ |
| generalizes | factorial ⓘ |
| hasFormula | (a)_n = \frac{\Gamma(a+n)}{\Gamma(a)} when a is not a nonpositive integer and n is a nonnegative integer ⓘ |
| hasInverseRelation | (a)_n = (-1)^n (-a)_n^{\downarrow} where ^{\downarrow} denotes falling factorial ⓘ |
| hasProperty |
holomorphic in a where Gamma function is defined
ⓘ
multiplicative over disjoint index ranges ⓘ polynomial in a of degree n for fixed n ⓘ |
| introducedBy | Leo Pochhammer ⓘ |
| namedAfter | Leo Pochhammer ⓘ |
| notationType | compact product notation ⓘ |
| recurrenceRelation |
(a)_n = a(a+1)_{n-1}
ⓘ
(a)_{n+1} = (a+n)(a)_n ⓘ |
| relatedTo |
Gamma function
ⓘ
falling factorial ⓘ |
| specialCase |
(1)_n = n!
ⓘ
(a)_1 = a ⓘ |
| usedFor | rising factorial ⓘ |
| usedIn |
Beta function identities
ⓘ
Gamma function identities ⓘ
linked to:
Gamma function
Gauss hypergeometric function ⓘ confluent hypergeometric function ⓘ generalized hypergeometric series ⓘ hypergeometric functions ⓘ orthogonal polynomials ⓘ power series ⓘ series expansions ⓘ special functions ⓘ |
| usedToExpress |
binomial coefficients in generalized form
ⓘ
coefficients of hypergeometric series ⓘ moments in probability distributions ⓘ terms of orthogonal polynomial sequences ⓘ |
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: Pochhammer symbol Description of subject: The Pochhammer symbol is a mathematical notation representing rising factorials, widely used in series expansions, special functions, and hypergeometric functions.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Pochhammer symbol
linked to: Pochhammer symbol