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.
Description
A Gaussian or q-binomial coefficient is a polynomial analogue of a binomial coefficient.
A Gaussian or q-binomial coefficient is a polynomial analogue of a binomial coefficient. The coefficients encode the q dependence and sum to the ordinary binomial coefficient at q=1.1
The tool returns coefficients. The indexing and output conventions below determine how to read these values.
When to use q-Binomial Coefficient Polynomial
Use this to obtain the entire Gaussian binomial polynomial for a chosen n and k, inspect coefficients by degree, or compare its q=1 value with an ordinary binomial coefficient. No numeric q input is required.
How to use q-Binomial Coefficient 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, k 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 | Upper q-binomial index (resource limit 64) | integer; minimum 0; maximum 64 | Required |
| k | Lower q-binomial index (resource limit 64) | integer; minimum 0; maximum 64 | Required |
Use k↔n−k symmetry and descending dynamic-programming rows, adding the shifted preceding polynomial. Return exact coefficient strings in ascending degree order.
[n choose k]_q=[n−1 choose k]_q+q^(n−k)[n−1 choose k−1]_q The symbols and boundary conditions refer to the supplied fields and the implementation described above.
| Output field | Meaning | Value type |
|---|---|---|
| coefficients | Gaussian polynomial coefficients from degree zero upward | list of string (integer/arbitrary) |
{
"input": {
"n": 4,
"k": 2
},
"output": {
"coefficients": [
"1",
"1",
"2",
"1",
"1"
]
}
}Limitations and assumptions
No numeric q is evaluated. For k>n return the zero polynomial ["0"]; both indexes are capped at 64.
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-Factorial Polynomial when you need its operation: 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. Choose Pascal Triangle Row for the complementary operation: Return the zero-indexed nth row of Pascal’s triangle as exact decimal binomial coefficients; n is capped at 1,000 to bound the output.
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-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.
- Pascal Triangle Row
Return the zero-indexed nth row of Pascal’s triangle as exact decimal binomial coefficients; n is capped at 1,000 to bound the output.