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.

Integer-order Bessel series

Definition

For an integer $n \geq 0$ and $z \in C$ , define

$J_{n} (z) = \sum_{k = 0}^{\infty} \frac{(- 1)^{k} (z / 2)^{2 k + n}}{k! (k + n)!}, I_{n} (z) = \sum_{k = 0}^{\infty} \frac{(z / 2)^{2 k + n}}{k! (k + n)!} .$

These are the integer-order Bessel and modified Bessel functions used on this page. For real $x \geq 0$ , the scaled modified Bessel value means $I_{n}^{s c} (x) = e^{- x} I_{n} (x)$ . This notation does not assert a finite evaluation method or a truncation bound; those are proved separately.

Used by

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

DLMF, §10.2(ii), equation 10.2.2 (standard reference, not scraped)
DLMF, §10.25(ii), equation 10.25.2 (standard reference, not scraped)
N. M. Temme, “Numerical aspects of special functions,” Acta Numerica 17 (2008), §§2 and 4 (standard reference, not scraped)