CorollaryStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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.

The elementary recurrences for the real Beta function

Statement

For $p, q > 0$ ,

$B (p + 1, q) = \frac{p}{p + q} B (p, q), B (p, q + 1) = \frac{q}{p + q} B (p, q),$

and consequently $B (p, q) = B (p + 1, q) + B (p, q + 1)$ .

Facts & Assumptions

Given: Positive real parameters $p, q$ .

[F1]

For $p, q > 0$ , $B (p, q) = Γ (p) Γ (q) / Γ (p + q)$ (The real Beta--Gamma identity).

[F2]

For every $s > 0$ , $Γ (s + 1) = s Γ (s)$ (The real Gamma functional equation $Γ (s + 1) = s Γ (s)$ ).

Proof

technique · direct

1.1

By [F1] and [F2], $B (p + 1, q) = Γ (p + 1) Γ (q) / Γ (p + q + 1) = p B (p, q) / (p + q)$ .

F1F2algebra

1.2

Similarly, $B (p, q + 1) = q B (p, q) / (p + q)$ .

F1F2algebra

2.1

Adding steps 1.1 and 1.2 and using $p + q > 0$ gives $B (p + 1, q) + B (p, q + 1) = B (p, q)$ .

step 1.1step 1.2algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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

University of Toronto MAT237Y1, The Gamma Function and the Beta Function, §2.2 (standard reference, not scraped)