{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:47:35Z","timestamp":1760060855316,"version":"build-2065373602"},"reference-count":44,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T00:00:00Z","timestamp":1758758400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Fundamental Research Funds for the Provincial Universities of Liaoning, China","award":["LJ212410150007"],"award-info":[{"award-number":["LJ212410150007"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>By means of the Lagrange interpolation, we derive two trigonometric identities that are utilized to evaluate, in closed forms, eight classes of power sums of trigonometric functions over equally distributed angles around the unit circle. When the mth term is removed from the sums, four classes (among eight) are shown to admit also closed expressions, that are surprisingly independent of the integer parameter \u201cm\u201d.<\/jats:p>","DOI":"10.3390\/axioms14100724","type":"journal-article","created":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T12:48:28Z","timestamp":1758804508000},"page":"724","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Trigonometric Sums via Lagrange Interpolation"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1972-4482","authenticated-orcid":false,"given":"Marta Na","family":"Chen","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Zhoukou Normal University, Zhoukou 466001, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8425-212X","authenticated-orcid":false,"given":"Wenchang","family":"Chu","sequence":"additional","affiliation":[{"name":"Independent Researcher, Via Dalmazio Birago 9\/E, 73100 Lecce, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0007-4200-9272","authenticated-orcid":false,"given":"Xiaoyuan","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Science, Dalian Jiaotong University, Dalian 116028, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1017\/S001309150002383X","article-title":"Some integer\u2013valued trigonometric sums","volume":"40","author":"Byrne","year":"1997","journal-title":"Proc. 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