q-Factorial Polynomial
Return exact coefficients of [n]_q!=∏(j=1…n)(1+q+…+q^(j−1)) in ascending degree order, including the empty product one and a 64-index resource limit.
Description
A q-factorial is a polynomial formed by multiplying q-integers.
A q-factorial is a polynomial formed by multiplying q-integers. Its coefficients preserve the symbolic q dependence, unlike evaluating an ordinary factorial at a number.1
The tool returns coefficients. The indexing and output conventions below determine how to read these values.
When to use q-Factorial Polynomial
Use this for the following task: Return exact coefficients of [n]_q!=∏(j=1…n)(1+q+…+q^(j−1)) in ascending degree order, including the empty product one and a 64-index resource limit. Use the input and output conventions below to reproduce the intended mathematical quantity.
How to use q-Factorial Polynomial and understand its results
- Choose the required indexes, integers or sequence entries. Preserve decimal-string encodings where the field table specifies strings.
- Enter n according to the field table, then run the tool.
- Read coefficients. Preserve exact decimal strings and the stated indexing or coefficient order.
- Keep exact decimal strings for subsequent integer arithmetic; distinguish them from any explicitly approximate numeric result.
| Input field | Meaning | Value type and bounds | Requirement |
|---|---|---|---|
| n | q-factorial index (resource limit 64) | integer; minimum 0; maximum 64 | Required |
Build exact BigInt coefficient arrays using a sliding-window multiplication for each factor. Return decimal-string coefficients from constant term to highest power.
[n]_q! = Π[j=1..n](1+q+…+q^(j−1)) The symbols and boundary conditions refer to the supplied fields and the implementation described above.
| Output field | Meaning | Value type |
|---|---|---|
| coefficients | q-factorial coefficients from degree zero upward | list of string (integer/arbitrary) |
{
"input": {
"n": 3
},
"output": {
"coefficients": [
"1",
"2",
"2",
"1"
]
}
}Limitations and assumptions
No numeric q is accepted. Index zero returns ["1"], and the index is capped at 64 to bound polynomial degree and coefficient growth.
Use the declared integer bounds and string formats. Large exact outputs remain decimal strings; numerical approximations use ordinary floating-point arithmetic. A rejected input is different from a valid zero count or false membership flag.
Alternative or Complementary methods
Choose q-Binomial Coefficient Polynomial when you need its operation: Return exact coefficients of the Gaussian binomial [n choose k]_q in ascending degree order, with zero for k>n and indices capped at 64. Choose Multinomial Coefficient for the complementary operation: Count arrangements with the supplied nonnegative category counts using exact arithmetic; at most 128 categories and 2,000 total elements are admitted.
Background reference: Digital Library of Mathematical Functions, §17.2.
References
-
Digital Library of Mathematical Functions, §17.2 — National Institute of Standards and Technology
Similar or alternative tools
- q-Binomial Coefficient Polynomial
Return exact coefficients of the Gaussian binomial [n choose k]_q in ascending degree order, with zero for k>n and indices capped at 64.
- Multinomial Coefficient
Count arrangements with the supplied nonnegative category counts using exact arithmetic; at most 128 categories and 2,000 total elements are admitted.