Partition Generating Function Approximation
Evaluate truncated real generating product ∏_(k=1..N)(1−q^k)^−1 with stable log accumulation, not exact partition coefficient counting or the exponent-sequence transform.
Description
The ordinary partition generating function can be represented as a product of reciprocal factors.
The ordinary partition generating function can be represented as a product of reciprocal factors. A finite factor count evaluates a truncated numeric product rather than extracting partition coefficients.1
Evaluate truncated real generating product ∏_(k=1..N)(1−q^k)^−1 with stable log accumulation, not exact partition coefficient counting or the exponent-sequence transform.
When to use Partition Generating Function Approximation
Use this tool to evaluate truncated real generating product ∏_(k=1..N)(1−q^k)^−1 with stable log accumulation, not exact partition coefficient counting or the exponent-sequence transform.
Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
How to use Partition Generating Function Approximation and understand its results
- Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
- Enter realArgumentQ, productFactors using the field table and calculation rules, then run the tool.
- Read finitePartitionGeneratingProduct, logFinitePartitionGeneratingProduct. The explanation below defines the units, signs, ordering and any flags or placeholder values.
- If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.
| Input field | Meaning | Value type; see rules below | Requirement |
|---|---|---|---|
| realArgumentQ | Real argument q | number | Required |
| productFactors | Product factors | number | Required |
Enter real q in [0,1) and an integer factor count 1–100,000. Accumulate −log1p(−q^k), return that log and its exponential; q=0 gives product 1 and log 0.
finite product = ∏[k=1..N](1−q^k)⁻¹ Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.
| Output field | Meaning | Value type |
|---|---|---|
| finitePartitionGeneratingProduct | Finite partition generating product | number |
| logFinitePartitionGeneratingProduct | Log finite partition generating product | number |
{
"input": {
"realArgumentQ": 0.1,
"productFactors": 3
},
"output": {
"finitePartitionGeneratingProduct": 1.1234579135813705,
"logFinitePartitionGeneratingProduct": 0.11641135184491129
}
}Limitations and assumptions
No omitted-factor bound is supplied. A nonfinite product is rejected even when its logarithm is representable.
Numeric inputs and returned numeric values must be finite; an unrepresentable returned value produces an error. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.
Alternative or Complementary methods
Use Partition Transform Calculator: Compute exact coefficients through N of formal product ∏(1−x^k)^(-a_k), allowing signed caller exponent sequence; not ordinary unrestricted partition count or finite q product evaluation. For a complementary task, use Dedekind Eta Approximation: Approximate real positive-nome Dedekind eta q^(1/24)∏_(n=1..N)(1−q^n), τ purely imaginary and q=exp(2πiτ); explicit finite product, not generic complex eta or a duplicate alias.
References
Similar or alternative tools
- Partition Transform Calculator
Compute exact coefficients through N of formal product ∏(1−x^k)^(-a_k), allowing signed caller exponent sequence; not ordinary unrestricted partition count or finite q product evaluation.
- Dedekind Eta Approximation
Approximate real positive-nome Dedekind eta q^(1/24)∏_(n=1..N)(1−q^n), τ purely imaginary and q=exp(2πiτ); explicit finite product, not generic complex eta or a duplicate alias.