PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-16

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Rational formal power series are closed under sums and Cauchy products

Statement

Let $R$ be a commutative ring. If $F, G \in R ⟦ x ⟧$ are rational, then both $F + G$ and the Cauchy product $F G$ are rational. This includes zero series and polynomial presentations.

Facts & Assumptions

Given: Rational presentations $F = P_{1} / Q_{1}$ and $G = P_{2} / Q_{2}$ over a commutative ring $R$ .

[L1]

A rational formal series has a presentation $P / Q$ whose denominator has unit constant coefficient (Rational formal power series, proper presentations and reduced denominators).

[L2]

A formal series is a unit exactly when its constant coefficient is a unit (A formal power series is a unit exactly when its constant coefficient is a unit).

Proof

technique · direct

1.1

The product $Q_{1} Q_{2}$ has unit constant coefficient $Q_{1} (0) Q_{2} (0)$ , so it is a unit of $R ⟦ x ⟧$ by [L2].

givenL1L2

2.1

The identities $F + G = (P_{1} Q_{2} + P_{2} Q_{1}) / (Q_{1} Q_{2})$ and $F G = P_{1} P_{2} / (Q_{1} Q_{2})$ have polynomial numerators and the denominator from step 1.1, so [L1] makes both series rational. The formulas remain valid when a numerator is zero or a denominator is $1$ .

step 1.1L1algebra∎

Depends on

Used by

Words over a finite alphabet avoiding finitely many nonempty factors have a rational length generating function Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 20 results over 12 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

R. P. Stanley, Enumerative Combinatorics, vol. 1, 2nd ed., Section 4.2 (standard reference, not scraped)