CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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$d$ initial values determine a sequence satisfying a fixed order- $d$ recurrence

Statement

Fix a field $K$ , an integer $d \geq 1$ , and coefficients $c_{1}, \dots, c_{d} \in K$ with $c_{d} \neq 0$ . For every initial list $(u_{0}, \dots, u_{d - 1}) \in K^{d}$ , there is exactly one sequence $(a_{n})_{n \geq 0}$ satisfying

$a_{n + d} + c_{1} a_{n + d - 1} + \dots + c_{d} a_{n} = 0 (n \geq 0)$

and $a_{i} = u_{i}$ for $i < d$ .

Facts & Assumptions

Given: A field $K$ , a positive order $d$ , fixed recurrence coefficients with nonzero trailing coefficient, and an initial list in $K^{d}$ .

[L1]

Initial-value extraction is a linear isomorphism from the fixed-recurrence sequence space to $K^{d}$ (The initial-value, recurrence-sequence, numerator and fixed-denominator rational-series spaces all have dimension $d$ ).

Proof

technique · direct

1.1

Surjectivity in [L1] gives a recurrence sequence with the prescribed $d$ initial values.

givenL1

2.1

Injectivity in [L1] says that two recurrence sequences with the same initial list are equal, proving uniqueness. This includes $d = 1$ , where one initial value determines every later term.

step 1.1L1∎

Depends on

The initial-value, recurrence-sequence, numerator and fixed-denominator rational-series spaces all have dimension $d$

Used by

FALSE: The recurrence coefficients alone determine the sequence False statement

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 27 results over 10 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

M. Waldschmidt, Linear Recurrence Sequences VI, slides 5-6 (standard reference, not scraped)

dd initial values determine a sequence satisfying a fixed order-dd recurrence