hasGeneratingFunction
P46242
predicate
Indicates that one entity serves as the generating function associated with, or defining, another entity.
All labels observed (6)
| Label | Occurrences |
|---|---|
| generatingFunction | 6 |
| hasGeneratingFunction canonical | 3 |
| haveGeneratingFunction | 2 |
| generatingFunctionType | 1 |
| givesGeneratingFunction | 1 |
| usesGeneratingFunction | 1 |
Description generation (PDg)
The one-sentence description above was generated by prompting gpt-5.1 with the predicate name and this instruction.
Instruction
Given a predicate that represents a relationship or action between entities, generate a one-sentence description explaining its meaning. # Instructions Focus on describing the relationship, not the entities themselves. # Response Format Begin the description with \' Indicates...\'
Input
Predicate: hasGeneratingFunction
Generated description
Indicates that one entity serves as the generating function associated with, or defining, another entity.
Sample triples (14)
| Subject | Object |
|---|---|
| Fibonacci sequence | x / (1 − x − x^2) ⓘ |
| Hardy–Ramanujan asymptotic formula | \sum_{n \ge 0} p(n) q^n = \prod_{m \ge 1} (1 - q^m)^{-1} via predicate surface "usesGeneratingFunction" ⓘ |
| Bernoulli numbers | t/(e^t - 1) via predicate surface "generatingFunction" ⓘ |
| Jacobi polynomials | known closed-form generating function in t ⓘ |
| Jacobi’s four-square theorem | (θ_3(q))^4 = 1 + ∑_{n≥1} r_4(n) q^n via predicate surface "givesGeneratingFunction" ⓘ |
| Jordan’s totient functions | Dirichlet generating function via predicate surface "generatingFunctionType" ⓘ |
| Ramanujan tau function | Δ(z) = q ∏_{n≥1} (1 - q^n)^{24} = ∑_{n≥1} τ(n) q^n with q = e^{2πiz} via predicate surface "generatingFunction" ⓘ |
| Bernoulli polynomials | t e^{xt} / (e^t - 1) = Σ_{n=0}^∞ B_n(x) t^n / n! via predicate surface "generatingFunction" ⓘ |
| Catalan numbers | C(x) = (1 - sqrt(1-4x)) / (2x) via predicate surface "generatingFunction" ⓘ |
| OEIS A002849 | Product_{k>=0} (1 + x^(2k+1)) ⓘ |
| Liouville function | Dirichlet generating function is ζ(2s)/ζ(s) via predicate surface "generatingFunction" ⓘ |
| Legendre polynomials | 1/sqrt(1-2xt+t^2) = sum_{n=0}^∞ P_n(x) t^n via predicate surface "haveGeneratingFunction" ⓘ |
| Chebyshev polynomials of the first kind | \sum_{n=0}^{\infty} T_n(x) t^n = \frac{1 - xt}{1 - 2xt + t^2} via predicate surface "generatingFunction" ⓘ |
| Gegenbauer polynomials | (1-2xt+t^2)^{-\lambda} = \sum_{n=0}^{\infty} C_n^{(\lambda)}(x) t^n via predicate surface "haveGeneratingFunction" ⓘ |