{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,11]],"date-time":"2026-04-11T02:23:39Z","timestamp":1775874219090,"version":"3.50.1"},"reference-count":18,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,8,12]],"date-time":"2022-08-12T00:00:00Z","timestamp":1660262400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The main aim of this study is to define parametric kinds of \u03bb-Array-type polynomials by using q-trigonometric polynomials and to investigate some of their analytical properties and applications. For this purpose, many formulas and relations for these polynomials, including some implicit summation formulas, differentiation rules, and relations with the earlier polynomials by utilizing some series manipulation method are derived. Additionally, as an application, the zero values of q-Array-type polynomials are presented by the tables and multifarious graphical representations for these zero values are drawn.<\/jats:p>","DOI":"10.3390\/sym14081675","type":"journal-article","created":{"date-parts":[[2022,8,15]],"date-time":"2022-08-15T23:44:03Z","timestamp":1660607043000},"page":"1675","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Explicit Properties of q-Cosine and q-Sine Array-Type Polynomials Containing Symmetric Structures"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5567-7444","authenticated-orcid":false,"given":"Maryam Salem","family":"Alatawi","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, University of Tabuk, Tabuk 71491, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4681-9885","authenticated-orcid":false,"given":"Waseem Ahmad","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, Al Khobar 31952, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Cheon Seoung","family":"Ryoo","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Hannam University, Daejeon 34430, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Alam, N., Khan, W.A., and Ryoo, C.S. (2022). A note on Bell-based Apostol-type Frobenius-Euler polynomials of complex variable with its certain applications. Mathematics, 10.","DOI":"10.3390\/math10122109"},{"key":"ref_2","first-page":"7172054","article-title":"Some identities of the multi poly-Bernoulli polynomials of complex variable","volume":"2021","author":"Muhiuddin","year":"2021","journal-title":"J. Funct. Spaces"},{"key":"ref_3","first-page":"3115424","article-title":"Construction on the degenerate poly-Frobenius-Euler polynomials of complex variable","volume":"2021","author":"Muhiuddin","year":"2021","journal-title":"J. Funct. Spaces"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1185","DOI":"10.1080\/09720502.2019.1709317","article-title":"Notes on q-Hermite based unified Apostol type polynomials","volume":"22","author":"Nisar","year":"2019","journal-title":"J. Interdiscip. 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Soc."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Kang, J.Y., and Ryoo, C.S. (2020). Various structures of the roots and explicit properties of q-cosine Bernoulli polynomials and q-sine Bernoulli polynomials. Mathematics, 8.","DOI":"10.3390\/math8040463"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"169348","DOI":"10.1155\/2012\/169348","article-title":"q-analogues of the Bernoulli and Genocchi polynomials and the Srivastava-Pinter addition theorems","volume":"2012","author":"Mahmudov","year":"2012","journal-title":"Discret. Dyn. Nat. Soc."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"108","DOI":"10.1186\/1687-1847-2013-108","article-title":"On a class of q-Bernoulli and q-Euler polynomials","volume":"2013","author":"Mahmudov","year":"2013","journal-title":"Adv. Diff. Equ."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Ryoo, C.S., and Kang, J.Y. (2020). Explicit properties of q-Cosine and q-Sine Euler polynomials containing symmetric structures. Symmetry, 12.","DOI":"10.3390\/sym12081247"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"37","DOI":"10.5937\/MatMor2201037M","article-title":"A generalization of the array type polynomials","volume":"26","author":"Jamei","year":"2022","journal-title":"Math. Moravica"},{"key":"ref_18","first-page":"5702","article-title":"Some generalization of the Apostol-Genocchi polynomials and Stirling numbers of the second kind","volume":"217","author":"Luo","year":"2011","journal-title":"Appl. Math. Comput."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/8\/1675\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:07:42Z","timestamp":1760141262000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/8\/1675"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,12]]},"references-count":18,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2022,8]]}},"alternative-id":["sym14081675"],"URL":"https:\/\/doi.org\/10.3390\/sym14081675","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,8,12]]}}}