DefinitionDefinition: AI-adaptedProof: 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.

Complementary solution for the integer Bessel recurrence

Definition

Fix real $x > 0$ . Let $D_{0} (x) = - J_{1} (x)$ and $D_{1} (x) = J_{0} (x)$ , and for every integer $j \geq 1$ define

$D_{j + 1} (x) = \frac{2 j}{x} D_{j} (x) - D_{j - 1} (x) .$

Thus $D$ is the second sequence obtained from the same order recurrence as $J_{j} (x)$ . It is a local auxiliary solution, not the Bessel function of the second kind.

Depends on

Integer-order Bessel series

Used by

Dependency tree · one level

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Sources

DLMF, §3.6(iii), Miller's Algorithm (standard reference, not scraped)
N. M. Temme, “Numerical aspects of special functions,” Acta Numerica 17 (2008), §§2 and 4 (standard reference, not scraped)