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Article ID: CBB311764284

Analytic and Arithmetic Methods in Liouville's Identities (2020)

Enea, Maria Rosaria (Author)
Ferraro, Giovanni (Author)

Historia Mathematica
Volume: 53
Pages: 48-70

DOI: 10.1016/j.hm.2020.06.003

URI: https://www.sciencedirect.com/science/article/pii/S031508602030046X

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The history of Liouville's identities is of remarkable interest especially as far as the methods used to prove them are concerned. In Liouville's opinion, the identities had to be proved by merely arithmetic considerations. Some mathematicians, such as Uspensky, developed Liouville's approach and used elementary methods. However, other mathematicians (among them Nazimoff and Bell) followed Hermite's suggestions of deriving Liouville's identities from the theory of elliptic functions: most of them thought that analytic methods were preferable since they allowed one to discover new identities (and not only to prove known ones). Sommario La storia delle identità di Liouville è di notevole interesse soprattutto per quanto riguarda i metodi usati per la loro dimostrazione. Secondo Liouville, le identità dovevano essere dimostrate attraverso considerazioni puramente aritmetiche. Alcuni matematici, come Uspensky, svilupparono l'approccio di Liouville usando metodi elementari. Tuttavia, altri matematici (tra cui Nazimoff e Bell) seguirono i suggerimenti di Hermite ottenendo le identità di Liouville dalla teoria delle funzioni ellittiche: la maggior parte di loro pensava che i metodi analitici fossero preferibili poiché permettevano di scoprire nuove identità (e non solo di dimostrare quelle già note).

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