Triple
T16876412
| Position | Surface form | Disambiguated ID | Type / Status |
|---|---|---|---|
| Subject | Finite Operator Calculus |
E421310
|
entity |
| Predicate | relatedTo |
P37
|
FINISHED |
| Object |
Sheffer sequences
Sheffer sequences are families of polynomial sequences characterized by specific generating functions and shift-invariance properties, playing a central role in the theory of finite operator calculus and umbral calculus.
|
E1237909
|
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: Sheffer sequences | Statement: [Finite Operator Calculus, relatedTo, Sheffer sequences]
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Sheffer sequences Context triple: [Finite Operator Calculus, relatedTo, Sheffer sequences]
-
A.
Bernoulli polynomials
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.
-
B.
Jack polynomials
Jack polynomials are a family of symmetric polynomials depending on a continuous parameter that generalize several classical symmetric functions and play a key role in algebraic combinatorics, representation theory, and mathematical physics.
-
C.
Christoffel–Darboux formula
The Christoffel–Darboux formula is a key result in the theory of orthogonal polynomials that provides an explicit expression for sums of products of such polynomials, with important applications in approximation theory and mathematical physics.
-
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.
-
E.
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.
- 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: Sheffer sequences Triple: [Finite Operator Calculus, relatedTo, Sheffer sequences]
Generated description
Sheffer sequences are families of polynomial sequences characterized by specific generating functions and shift-invariance properties, playing a central role in the theory of finite operator calculus and umbral calculus.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Sheffer sequences Target entity description: Sheffer sequences are families of polynomial sequences characterized by specific generating functions and shift-invariance properties, playing a central role in the theory of finite operator calculus and umbral calculus.
-
A.
Bernoulli polynomials
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.
-
B.
Jack polynomials
Jack polynomials are a family of symmetric polynomials depending on a continuous parameter that generalize several classical symmetric functions and play a key role in algebraic combinatorics, representation theory, and mathematical physics.
-
C.
Christoffel–Darboux formula
The Christoffel–Darboux formula is a key result in the theory of orthogonal polynomials that provides an explicit expression for sums of products of such polynomials, with important applications in approximation theory and mathematical physics.
-
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.
-
E.
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.
- F. None of above. chosen
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_69d889d470fc8190b4aec199636c0c56 |
completed | April 10, 2026, 5:25 a.m. |
| NER | Named-entity recognition | batch_69e3b7f704a081909921d00b3c470472 |
completed | April 18, 2026, 4:57 p.m. |
| NED1 | Entity disambiguation (via context triple) | batch_6a00c2b4abd08190841c5bb0b0eaa177 |
completed | May 10, 2026, 5:39 p.m. |
| NEDg | Description generation | batch_6a00c355f4108190a4209599bf5f50da |
completed | May 10, 2026, 5:41 p.m. |
| NED2 | Entity disambiguation (via description) | batch_6a00c413314881909e308588af09ce2a |
completed | May 10, 2026, 5:44 p.m. |
Created at: April 10, 2026, 5:29 a.m.