{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,14]],"date-time":"2025-10-14T00:33:36Z","timestamp":1760402016455,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2020,5,2]],"date-time":"2020-05-02T00:00:00Z","timestamp":1588377600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11701447"],"award-info":[{"award-number":["11701447"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100007128","name":"Natural Science Foundation of Shaanxi Province","doi-asserted-by":"publisher","award":["2017JK1002"],"award-info":[{"award-number":["2017JK1002"]}],"id":[{"id":"10.13039\/501100007128","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we utilize the mathematical induction, the properties of symmetric polynomial sequences and Chebyshev polynomials to study the calculating problems of a certain reciprocal sums of Chebyshev polynomials, and give two interesting identities for them. These formulae not only reveal the close relationship between the trigonometric function and the Riemann \u03b6-function, but also generalized some existing results. At the same time, an error in an existing reference is corrected.<\/jats:p>","DOI":"10.3390\/sym12050704","type":"journal-article","created":{"date-parts":[[2020,5,5]],"date-time":"2020-05-05T06:41:20Z","timestamp":1588660880000},"page":"704","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["On the Chebyshev Polynomials and Some of Their Reciprocal Sums"],"prefix":"10.3390","volume":"12","author":[{"given":"Wenpeng","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Science, Xi\u2019an Technological University, Xi\u2019an 710021, China"},{"name":"School of Mathematics, Northwest University, Xi\u2019an 710127, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7982-9916","authenticated-orcid":false,"given":"Di","family":"Han","sequence":"additional","affiliation":[{"name":"School of Mathematics, Northwest University, Xi\u2019an 710127, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,5,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Borwein, P., and Erd\u00e9lyi, T. (1995). Polynomials and Polynomial Inequalities, Springer.","DOI":"10.1007\/978-1-4612-0793-1"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Li, X.X. (2015). Some identities involving Chebyshev polynomials. Math. Probl. Eng., 2015.","DOI":"10.1186\/s13662-015-0420-z"},{"key":"ref_3","first-page":"321","article-title":"Representing sums of finite products of Chebyshev polynomials of the second kind and Fibonacci polynomials in terms of Chebyshev polynomials","volume":"28","author":"Kim","year":"2018","journal-title":"Adv. Stud. Contemp. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1186\/s13662-019-2058-8","article-title":"Representation by several orthogonal polynomials for sums of finite products of Chebyshev polynomials of the first, third and fourth kinds","volume":"2019","author":"Kim","year":"2019","journal-title":"Adv. Differ. 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