Description
One formal power-series coefficient from Q(z)A(z)=P(z), with supplied prior coefficients aligned to Q tail coefficients.
A sequence rule specifies an index origin, state order, and relation among terms. Advancing one step, mapping an index, and checking a local residual answer different questions. Supplied coefficients or periods define a model; they do not establish that observations follow it.
One formal power-series coefficient from Q(z)A(z)=P(z), with supplied prior coefficients aligned to Q tail coefficients.
When to use this tool
Use Rational Generating Function Coefficient Step when you need this particular index map, recurrence step, count, or local algebraic term under supplied sequence conventions.
How to use the tool and how it works
- Enter numeratorCoefficient, denominatorTail, previousCoefficients, denominatorConstant. Keep parallel arrays aligned. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read coefficient. Use the output convention below.
Return coefficient=(numeratorCoefficient-dot(denominatorTail,previousCoefficients))/denominatorConstant. Require aligned arrays and nonzero constant.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| numeratorCoefficient | Numerator Coefficient | Required. Finite number. |
| denominatorTail | Denominator Tail | Required. Array of finite number; maximum items 256. |
| previousCoefficients | Previous Coefficients | Required. Array of finite number; maximum items 256. |
| denominatorConstant | Denominator Constant | Required. Finite number. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| coefficient | Coefficient | Required. Finite number. |
{
"input": {
"numeratorCoefficient": 0,
"denominatorTail": [
-1,
-1
],
"previousCoefficients": [
3,
2
],
"denominatorConstant": 1
},
"output": {
"coefficient": 5
}
}Limitations and assumptions
- Earlier coefficients must already be aligned in the intended convolution order. No generating expression is parsed, recurrence fitted, or other coefficient terms independently verified.
- The field tables state structural bounds; the mechanics also specify cross-field constraints. Malformed inputs and unsupported domains cause errors. A successful false flag, empty branch value, or zero placeholder has the meaning stated above and is distinct from an input error.
- Numeric calculations use binary floating point unless exact decimal-integer arithmetic is explicitly stated. No hidden measurement tolerance or decimal rounding is added; nonfinite or out-of-contract results are rejected. Signed zero is exported as zero.
Alternative or Complementary methods
Choose exact integer arithmetic when the index or coefficient range requires it. A complete sequence analysis needs actual terms and a consistent index and state-order convention beyond one local step.
References
Similar or alternative tools
- Linear Recurrence Coefficient Impulse Term
Return contribution=coefficients[index]*impulse for a valid zero-based index.
- Sequence: Linear Recurrence First Difference Step
Advance one short real linear recurrence with oldest-to-newest state and matching coefficient order, returning next state and its first difference; no symbolic recurrence solving.