{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T04:30:30Z","timestamp":1772253030867,"version":"3.50.1"},"reference-count":15,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2018,7,3]],"date-time":"2018-07-03T00:00:00Z","timestamp":1530576000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003725","name":"National Research Foundation of Korea","doi-asserted-by":"publisher","award":["2017R1A2B4006092"],"award-info":[{"award-number":["2017R1A2B4006092"]}],"id":[{"id":"10.13039\/501100003725","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Here, we consider the sums of finite products of Chebyshev polynomials of the third and fourth kinds. Then, we represent each of those sums of finite products as linear combinations of the four kinds of Chebyshev polynomials, which involve the hypergeometric function 3F2.<\/jats:p>","DOI":"10.3390\/sym10070258","type":"journal-article","created":{"date-parts":[[2018,7,3]],"date-time":"2018-07-03T11:12:58Z","timestamp":1530616378000},"page":"258","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Representing Sums of Finite Products of Chebyshev Polynomials of Third and Fourth Kinds by Chebyshev Polynomials"],"prefix":"10.3390","volume":"10","author":[{"given":"Taekyun","family":"Kim","sequence":"first","affiliation":[{"name":"Department of Mathematics, Kwangwoon University, Seoul 139-701, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9599-7015","authenticated-orcid":false,"given":"Dae San","family":"Kim","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Sogang University, Seoul 121-742, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dolgy Dmitriy","family":"Victorovich","sequence":"additional","affiliation":[{"name":"Institute of Natural Sciences, Far Eastern Federal University, 690950 Vladivostok, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Cheon Seoung","family":"Ryoo","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Hannam University, Daejeon 306-791, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,7,3]]},"reference":[{"key":"ref_1","unstructured":"Andrews, G.E., Askey, R., and Roy, R. (1999). Special Functions, Cambridge University Press. Encyclopedia of Mathematics and Its Applications 71."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Beals, R., and Wong, R. (2016). Special Functions and Orthogonal Polynomials, Cambridge University Press. Cambridge Studies in Advanced Mathematics 153.","DOI":"10.1017\/CBO9781316227381"},{"key":"ref_3","unstructured":"Guo, D.R., and Xia, X.J. (1989). Special Functions, World Scientific Publishing Co., Inc."},{"key":"ref_4","first-page":"3","article-title":"A temporally evolutionary equation in elasticity of micropolar bodies with voids","volume":"60","author":"Marin","year":"1998","journal-title":"Politehn. Univ. Buchar. Sci. Bull. Ser. A Appl. Math. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"33","DOI":"10.37193\/CJM.2013.01.12","article-title":"Weak solutions in elasticity of dipolar bodies with stretch","volume":"29","author":"Marin","year":"2013","journal-title":"Carpath. J. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1186\/s13661-016-0620-9","article-title":"On vibrations in thermoelasticity without energy dissipation for micropolar bodies","volume":"2016","author":"Marin","year":"2016","journal-title":"Bound. Value Probl."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Kim, T., Kim, D.S., Dolgy, D.V., and Kwon, J. (2018). Representing Sums of finite products of Chebyshev polynomials of the third and fourth kinds by Chebyshev Polynomials. Preprints, 2018060079.","DOI":"10.20944\/preprints201806.0079.v1"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1186\/s13662-017-1309-9","article-title":"Sums of finite products of Bernoulli functions","volume":"2017","author":"Agarwal","year":"2017","journal-title":"Adv. Differ. Equ."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Kim, T., Kim, D.S., Jang, G.-W., and Kwon, J. (2017). Sums of finite products of Euler functions. Advances in Real and Complex Analysis with Applications, Trends in Mathematics, Springer.","DOI":"10.1007\/978-981-10-4337-6_10"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1186\/s13662-017-1325-9","article-title":"Sums of finite products of Genocchi functions","volume":"2017","author":"Kim","year":"2017","journal-title":"Adv. Differ. Equ."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1016\/j.joems.2014.04.008","article-title":"On using third and fourth kinds Chebyshev polynomials for solving the integrated forms of high odd-order linear boundary value problems","volume":"23","author":"Doha","year":"2015","journal-title":"J. Egyptian Math. Soc."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1016\/j.jmaa.2013.03.009","article-title":"Application of a composition of generating functions for obtaining explicit formulas of polynomials","volume":"404","author":"Kruchinin","year":"2013","journal-title":"J. Math. Anal. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1016\/0377-0427(93)90148-5","article-title":"Chebyshev polynomials of the second, third and fourth kinds in approximation, indefinite integration, and integral transforms","volume":"49","author":"Mason","year":"1993","journal-title":"J. Comput. Appl. Math."},{"key":"ref_14","first-page":"172","article-title":"Some identities for Berounlli polynomials involving Chebyshev polynomials","volume":"16","author":"Kim","year":"2014","journal-title":"J. Comput. Anal. Appl."},{"key":"ref_15","first-page":"361","article-title":"Identities involving Bernoulli and Euler polynomials arising from Chebyshev polynomials","volume":"15","author":"Kim","year":"2012","journal-title":"Proc. Jangjeon Math. Soc."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/10\/7\/258\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:11:07Z","timestamp":1760195467000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/10\/7\/258"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,7,3]]},"references-count":15,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2018,7]]}},"alternative-id":["sym10070258"],"URL":"https:\/\/doi.org\/10.3390\/sym10070258","relation":{"has-preprint":[{"id-type":"doi","id":"10.20944\/preprints201806.0079.v1","asserted-by":"object"}]},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,7,3]]}}}