DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-27

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.
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Exponential generating functions over a commutative $Q$ -algebra

Definition

Let $R$ be a commutative $Q$ -algebra and let $(a_{n})_{n \geq 0}$ be a sequence in $R$ . Its exponential generating function is the formal series

$A (x) : = \sum_{n \geq 0} a_{n} \frac{x^{n}}{n!} \in R ⟦ x ⟧,$

formed inside Formal power series over a commutative ring and the coefficient-extraction functional $[x^{n}]$ .

The $Q$ -algebra structure is part of the definition: it is what makes the scalars $1 / n!$ meaningful in $R$ . The resulting series is formal rather than analytic, and the formal $\exp$ and $\log$ used later are those of Formal exponential, logarithm, and binomial powers over a commutative $Q$ -algebra.

Depends on

Used by

The labelled constructions translate into the usual exponential-generating-function rules Theorem

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Herbert S. Wilf, generatingfunctionology, 2nd ed., §2.3 (standard reference, not scraped)
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics — Symbolic Combinatorics (standard reference, not scraped)

Exponential generating functions over a commutative QQ-algebra