{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,2]],"date-time":"2026-01-02T07:06:33Z","timestamp":1767337593962},"reference-count":26,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2017,6,7]],"date-time":"2017-06-07T00:00:00Z","timestamp":1496793600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Struct. Comp. Sci."],"published-print":{"date-parts":[[2019,1]]},"abstract":"<jats:p>We present a comparative study of four product operators on weighted languages: (i) the <jats:italic>convolution<\/jats:italic>, (ii) the <jats:italic>shuffle<\/jats:italic>, (iii) the <jats:italic>infiltration<\/jats:italic> and (iv) the <jats:italic>Hadamard<\/jats:italic> product. Exploiting the fact that the set of weighted languages is a final coalgebra, we use coinduction to prove that an operator of the classical difference calculus, the <jats:italic>Newton transform<\/jats:italic>, generalises from infinite sequences to weighted languages. We show that the Newton transform is an isomorphism of rings that transforms the Hadamard product of two weighted languages into their infiltration product, and we develop various representations for the Newton transform of a language, together with concrete calculation rules for computing them.<\/jats:p>","DOI":"10.1017\/s0960129517000159","type":"journal-article","created":{"date-parts":[[2017,6,7]],"date-time":"2017-06-07T08:38:55Z","timestamp":1496824735000},"page":"38-66","source":"Crossref","is-referenced-by-count":3,"title":["Newton series, coinductively: a comparative study of composition"],"prefix":"10.1017","volume":"29","author":[{"given":"HENNING","family":"BASOLD","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"HELLE HVID","family":"HANSEN","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"JEAN-\u00c9RIC","family":"PIN","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"JAN","family":"RUTTEN","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2017,6,7]]},"reference":[{"key":"S0960129517000159_ref10","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-010-2196-8"},{"key":"S0960129517000159_ref18","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-20028-6_8"},{"key":"S0960129517000159_ref6","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-73235-5"},{"key":"S0960129517000159_ref13","volume-title":"Concrete Mathematics","author":"Graham","year":"1994"},{"key":"S0960129517000159_ref19","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejc.2013.09.009"},{"key":"S0960129517000159_ref24","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(02)00895-2"},{"key":"S0960129517000159_ref12","volume-title":"Automata, Languages and Machines (Vol. A)","author":"Eilenberg","year":"1974"},{"key":"S0960129517000159_ref3","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-25150-9_7"},{"key":"S0960129517000159_ref2","unstructured":"Bartels F. 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