Alphabeta Math
CounterexampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-08-30
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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.

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A periodic perturbation preserves the Gamma recurrence and factorial values

Statement refuted

Every meromorphic function with the Gamma recurrence and the Gamma factorial values must equal Gamma.

Facts & Assumptions

Given: The perturbed function F(z)=Γ(z)esin⁡(2πz).

[L1]

The false-statement refutation already proves that F satisfies the Gamma recurrence and the Gamma factorial values, but is not equal to Gamma (FALSE: the Gamma recurrence and factorial values characterize Gamma).

Counterexample

technique · direct
1.1

The function F(z)=Γ(z)esin⁡(2πz) is meromorphic, because Gamma is meromorphic and the exponential factor is entire.

given
2.1

By [L1], this same F has the Gamma recurrence and the Gamma factorial values while still differing from Gamma. Hence it is a counterexample to the statement refuted.

step 1.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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