Triple

T7194299
Position Surface form Disambiguated ID Type / Status
Subject Vandermonde's identity E167770 entity
Predicate generalizesTo P57966 FINISHED
Object Chu–Vandermonde identity
The Chu–Vandermonde identity is a classical hypergeometric summation formula that extends Vandermonde's combinatorial identity to generalized binomial coefficients and rising factorials.
E167770 NE FINISHED

How this triple was built (4 steps)

Every LLM step that produced this triple, in pipeline order — named-entity classification, the disambiguation choices (the exact options shown, with the pick highlighted), and the generated description. The batch + timestamp of each is in the Provenance table below.

NER Named-entity recognition gpt-5-mini
Instruction
Given a phrase, classify it is english named entity (e.g., persons, organizations, works of art) in Latin script, or not (e.g., literals, dates, URLs, verbose phrases). For disambiguation, the statement where the phrase occurs as object is also given. Please return a JSON object with `phrase` (string, the phrase being analyzed) and `is_ne` (boolean, indicating whether the phrase is a Named Entity).
Input
Phrase: Chu–Vandermonde identity | Statement: [Vandermonde's identity, generalizesTo, Chu–Vandermonde identity]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Chu–Vandermonde identity
Context triple: [Vandermonde's identity, generalizesTo, Chu–Vandermonde identity]
  • A. Vandermonde's identity
    Vandermonde's identity is a fundamental combinatorial formula that expresses a binomial coefficient with a sum index as a sum of products of binomial coefficients, often visualized via counting arguments or generating functions.
  • B. Pascal's identity
    Pascal's identity is a fundamental combinatorial formula that relates adjacent binomial coefficients and underlies many proofs and properties of binomial expansions.
  • C. 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.
  • D. Vandermonde matrix
    A Vandermonde matrix is a structured matrix whose rows (or columns) are geometric progressions of given numbers, widely used in polynomial interpolation, determinant theory, and numerical analysis.
  • E. Pochhammer symbol
    The Pochhammer symbol is a mathematical notation representing rising factorials, widely used in series expansions, special functions, and hypergeometric functions.
  • F. None of above. chosen
  • G. Unsure - the case is ambiguous/there is not enough information to decide.
NEDg Description generation gpt-5.1
Instruction
Generate a one-sentence description of the target entity. 
You are given a context triple in the form (subject, predicate, object), where the object is the target entity. 
# Instructions
Use the triple to infer relevant information about the entity. Describe the entity based on what is most defining, well-known. 
Avoid repeating the information from the triple, unless really essential.
# Response Format
Return only the sentence: "Description: [one-sentence description of the target entity]"
Input
Entity: Chu–Vandermonde identity
Triple: [Vandermonde's identity, generalizesTo, Chu–Vandermonde identity]
Generated description
The Chu–Vandermonde identity is a classical hypergeometric summation formula that extends Vandermonde's combinatorial identity to generalized binomial coefficients and rising factorials.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Chu–Vandermonde identity
Target entity description: The Chu–Vandermonde identity is a classical hypergeometric summation formula that extends Vandermonde's combinatorial identity to generalized binomial coefficients and rising factorials.
  • A. Vandermonde's identity chosen
    Vandermonde's identity is a fundamental combinatorial formula that expresses a binomial coefficient with a sum index as a sum of products of binomial coefficients, often visualized via counting arguments or generating functions.
  • B. Pascal's identity
    Pascal's identity is a fundamental combinatorial formula that relates adjacent binomial coefficients and underlies many proofs and properties of binomial expansions.
  • C. 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.
  • D. Vandermonde matrix
    A Vandermonde matrix is a structured matrix whose rows (or columns) are geometric progressions of given numbers, widely used in polynomial interpolation, determinant theory, and numerical analysis.
  • E. Pochhammer symbol
    The Pochhammer symbol is a mathematical notation representing rising factorials, widely used in series expansions, special functions, and hypergeometric functions.
  • F. None of above.

Provenance (5 batches)

The batch behind each pipeline step, in order, with when it ran. Timestamps are batch-level — stages were processed in waves, so the object chain (NER → NED1 → NEDg → NED2) reads in order, but predicate / elicitation batches can sit in a different wave.

Step Stage Batch ID Status When
creating Elicitation batch_69c6888b5248819090499a884ee3ec39 completed March 27, 2026, 1:39 p.m.
NER Named-entity recognition batch_69c6e9050164819081fd6a11d10f9833 completed March 27, 2026, 8:31 p.m.
NED1 Entity disambiguation (via context triple) batch_69c7bf9b8ff48190a561035f754922e9 completed March 28, 2026, 11:46 a.m.
NEDg Description generation batch_69c7c08e0db88190ac0142980288cd72 completed March 28, 2026, 11:50 a.m.
NED2 Entity disambiguation (via description) batch_69c7c1576818819087ad2d847f140433 completed March 28, 2026, 11:53 a.m.
Created at: March 27, 2026, 2:50 p.m.