{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,11,3]],"date-time":"2023-11-03T04:08:21Z","timestamp":1698984501382},"reference-count":19,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2021,4,29]],"date-time":"2021-04-29T00:00:00Z","timestamp":1619654400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,4,29]],"date-time":"2021-04-29T00:00:00Z","timestamp":1619654400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Period Math Hung"],"published-print":{"date-parts":[[2022,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>For each integer<jats:inline-formula><jats:alternatives><jats:tex-math>$$n\\ge 1$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>n<\/mml:mi><mml:mo>\u2265<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>we consider the unique polynomials<jats:inline-formula><jats:alternatives><jats:tex-math>$$P, Q\\in {\\mathbb {Q}}[x]$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>P<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>Q<\/mml:mi><mml:mo>\u2208<\/mml:mo><mml:mi>Q<\/mml:mi><mml:mo>[<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo>]<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>of smallest degree<jats:italic>n<\/jats:italic>that are solutions of the equation<jats:inline-formula><jats:alternatives><jats:tex-math>$$P(x)x^{n+1}+Q(x)(x+1)^{n+1}=1$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>P<\/mml:mi><mml:mrow><mml:mo>(<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:mrow><mml:msup><mml:mi>x<\/mml:mi><mml:mrow><mml:mi>n<\/mml:mi><mml:mo>+<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mo>+<\/mml:mo><mml:mi>Q<\/mml:mi><mml:mrow><mml:mo>(<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:mrow><mml:msup><mml:mrow><mml:mo>(<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo>+<\/mml:mo><mml:mn>1<\/mml:mn><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><mml:mo>+<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mo>=<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We derive numerous properties of these polynomials and their derivatives, including explicit expansions, differential equations, recurrence relations, generating functions, resultants, discriminants, and irreducibility results. We also consider some related polynomials and their properties.<\/jats:p>","DOI":"10.1007\/s10998-020-00376-5","type":"journal-article","created":{"date-parts":[[2021,4,29]],"date-time":"2021-04-29T21:39:12Z","timestamp":1619732352000},"page":"89-118","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Arithmetic properties of polynomial solutions of the Diophantine equation $$P(x)x^{n+1}+Q(x)(x+1)^{n+1}=1$$"],"prefix":"10.1007","volume":"84","author":[{"given":"Karl","family":"Dilcher","sequence":"first","affiliation":[]},{"given":"Maciej","family":"Ulas","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,4,29]]},"reference":[{"key":"376_CR1","first-page":"251","volume":"7","author":"M Davis","year":"1963","unstructured":"M. Davis, H. Putnam, Diophantine sets over polynomial rings. Ill. J. Math. 7, 251\u2013256 (1963)","journal-title":"Ill. J. 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