RemarkRemark: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Reciprocal-root convention: $χ (t) = \prod_{i} (t - λ_{i})^{m_{i}}$ corresponds to $Q (x) = \prod_{i} (1 - λ_{i} x)^{m_{i}}$

Statement

The recurrence convention of Constant-coefficient linear recurrences, their starting index and their characteristic polynomial pairs

$χ (t) = t^{d} + c_{1} t^{d - 1} + \dots + c_{d}$

with $Q (x) = 1 + c_{1} x + \dots + c_{d} x^{d}$ . Therefore, in any field over which $χ$ splits,

$χ (t) = \prod_{i} (t - λ_{i})^{m_{i}} ⟺ Q (x) = \prod_{i} (1 - λ_{i} x)^{m_{i}} .$

The roots of $Q$ are the reciprocals $λ_{i}^{- 1}$ , while the numbers appearing in recurrence closed forms are the characteristic roots $λ_{i}$ . The nonzero trailing coefficient in the recurrence makes every $λ_{i}$ nonzero. This convention is algebraic and does not assert convergence of $P / Q$ at any value of $x$ (Rational formal power series, proper presentations and reduced denominators).

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 11 results over 7 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

B. E. Sagan, Combinatorics: The Art of Counting, Theorem 3.7.1 (standard reference, not scraped)
M. Waldschmidt, Linear Recurrence Sequences VI, reciprocal characteristic polynomial (standard reference, not scraped)

Reciprocal-root convention: χ(t)=∏i(t−λi)miχ(t)=∏i​(t−λi​)mi​ corresponds to Q(x)=∏i(1−λix)miQ(x)=∏i​(1−λi​x)mi​