DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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.

Airy initial-value series

Definition

For $y (z) = \sum_{k \geq 0} c_{k} z^{k}$ , set $c_{2} = 0$ and $c_{k + 3} = c_{k} / ((k + 3) (k + 2))$ for $k \geq 0$ . Define $Ai$ by this series with

$c_{0} = \frac{1}{3^{2 / 3} Γ (2 / 3)}, c_{1} = - \frac{1}{3^{1 / 3} Γ (1 / 3)},$

and define $Bi$ by the same recurrence with

$c_{0} = \frac{1}{3^{1 / 6} Γ (2 / 3)}, c_{1} = \frac{3^{1 / 6}}{Γ (1 / 3)} .$

Here $Γ$ is Euler's integral on the right half-plane from Euler's Gamma function on the right half-plane. The standard Airy initial values are those in DLMF §9.2(ii); in particular, the power of $3$ in $Bi (0)$ is in the denominator.

Depends on

Euler's Gamma function on the right half-plane

Used by

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Sources

DLMF, §9.2(ii), equations 9.2.3–9.2.6 (standard reference, not scraped)
DLMF, §9.4, Maclaurin series (standard reference, not scraped)
N. M. Temme, “Numerical aspects of special functions,” Acta Numerica 17 (2008), §2 (standard reference, not scraped)