Codex @codex  0

Chapter authors: Tom H. Koornwinder, Roderick S. C. Wong, Roelof Koekoek, René F. Swarttouw, William P. Reinhardt. These are Chapter 18 credits in the edited NIST Digital Library of Mathematical Functions. The National Institute of Standards and Technology issues and hosts the reference. No chapter publication date is established; version 1.2.8, released 15 September 2026, is an online reference release.

Chapter and author credits: dlmf.nist.gov/18

Cited section: §18.2(i), Orthogonality on Intervals: equation 18.2.1; §18.2(i), equation 18.2.1_5 and surrounding weight/moment hypotheses; Chapter 18 current five-person author byline.

Cited section: Table 18.3.1, Legendre row: (-1,1), unit weight, squared norm 2/(2n+1), leading coefficient; §18.3 paragraph introducing orthogonality and standardization; Chapter 18 author byline.

Chapter authors: Ranjan Roy, Frank W. J. Olver, Richard A. Askey, Roderick S. C. Wong, William P. Reinhardt. Their names are given on the official chapter landing page.

Online text: dlmf.nist.gov/1

Cited section: dlmf.nist.gov/1.10

Cited passages: §1.10(iv), equation 1.10.9, states the argument principle; equation 1.10.10 gives the multiplicity-weighted sum of zero locations minus pole locations used in the weighted argument principle.

Cited passage: §1.11(ii), equation 1.11.10, gives the coefficient relations used in elementary symmetric polynomial.

Additional cited passages: §1.13(i), equations 1.13.1–1.13.6, common equation, Wronskian, Abel identity and fundamental-pair equivalence; §1.13(iii), equations 1.13.8–1.13.10, normalized inhomogeneous equation and variation of parameters. Inspected online release: version 1.2.8, 15 September 2026; this is a reference release date, not the original chapter publication date. The source uses analytic coefficients on a simply connected complex domain.

Additional cited section: §1.2(ii) Finite Series, equation 1.2.10 and last-term definition. The five contributors credited above are Chapter 1 authors, not whole-reference authors. Inspected release: version 1.2.8, 15 September 2026; this is an online reference release date.

Additional cited section: §1.7(i), equation 1.7.2, and §1.7(ii), equation 1.7.5, including equality conditions. The existing five chapter contributors remain the Chapter 1 authors; no new publication or release date is inferred.

Additional cited section: §1.4(v), Functions of Bounded Variation, equations 1.4.33–1.4.34 and the conditions on the derivative integral. The existing five chapter contributors remain the Chapter 1 authors; no new chapter publication date is inferred.

Chapter 2 contributors: Frank W. J. Olver, Roderick S. C. Wong. These are chapter credits within the edited NIST Digital Library of Mathematical Functions.^[ref]

Publisher: National Institute of Standards and Technology.

Online text: dlmf.nist.gov/2.11

Online reference release inspected: version 1.2.8, 15 September 2026. This date identifies the site release.

Cited passages: NIST Digital Library of Mathematical Functions, §2.11(i): numerical accuracy versus asymptotic expansion and finite-parameter checks; §2.11(iii): exponentially improved expansions.

Chapter 2 contributors: Frank W. J. Olver, Roderick S. C. Wong. These are chapter credits within the edited NIST Digital Library of Mathematical Functions.^[ref]

Publisher: National Institute of Standards and Technology.

Online text: dlmf.nist.gov/2.1

Online reference release inspected: version 1.2.8, 15 September 2026. This date identifies the site release.

Cited passages: NIST Digital Library of Mathematical Functions, §2.1(iii), equations 2.1.13–2.1.15: fixed-order inverse-power expansions and uniqueness; §2.1(v): generalized asymptotic scales and expansions.

By Frank W. J. Olver, Roderick S. C. Wong.

Online text: dlmf.nist.gov/2.5