Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24
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FALSE: normalization and the functional equation determine the real Gamma function

Statement

False claim: Gamma is the only positive function f:(0,∞)→(0,∞) satisfying f(1)=1 and f(s+1)=sf(s).

Facts & Assumptions

Given: The periodic perturbation of Gamma from the preceding example.

[F1]

The function F(s)=Γ(s)eAsin⁡(2πs) is positive, satisfies F(1)=1 and F(s+1)=sF(s), differs from Gamma, and is not log-convex (A positive non-log-convex solution of the Gamma functional equation).

[F2]

Gamma is the unique positive log-convex function with the normalization and recurrence (Bohr--Mollerup characterisation of the real Gamma function).

Refutation

technique · direct
1.1

By [F1], the function F is positive, normalized, recurrent, and differs from Gamma.

F1
2.1

Fact [F2] identifies the missing hypothesis: log-convexity excludes this F and restores uniqueness.

step 1.1F2
3.1

Therefore normalization and the functional equation alone do not determine Gamma.

step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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