{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T11:49:25Z","timestamp":1775476165327,"version":"3.50.1"},"reference-count":25,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2022,7,19]],"date-time":"2022-07-19T00:00:00Z","timestamp":1658188800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2022,7,19]],"date-time":"2022-07-19T00:00:00Z","timestamp":1658188800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2023,2]]},"DOI":"10.1007\/s12190-022-01774-3","type":"journal-article","created":{"date-parts":[[2022,7,19]],"date-time":"2022-07-19T12:06:48Z","timestamp":1658232408000},"page":"811-822","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Explicit formula for positive integer powers of k-tridiagonal Toeplitz matrices"],"prefix":"10.1007","volume":"69","author":[{"given":"Ji-Teng","family":"Jia","sequence":"first","affiliation":[]},{"given":"Jie","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Qi","family":"He","sequence":"additional","affiliation":[]},{"given":"Yu-Cong","family":"Yan","sequence":"additional","affiliation":[]},{"given":"Fatih","family":"Y\u0131lmaz","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,7,19]]},"reference":[{"issue":"1","key":"1774_CR1","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1137\/0706014","volume":"6","author":"CF Fischer","year":"1969","unstructured":"Fischer, C.F., Usmani, R.A.: Properties of some tridiagonal matrices and their application to boundary value problems. SIAM J. Numer. Anal. 6(1), 127\u2013142 (1969)","journal-title":"SIAM J. Numer. Anal."},{"key":"1774_CR2","volume-title":"Methods. Appl.","author":"SB Matrices","year":"1992","unstructured":"Matrices, S.B.: Methods. Appl. Oxford University Press, Boston (1992)"},{"issue":"1","key":"1774_CR3","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1016\/S0024-3795(01)00462-1","volume":"342","author":"WK Grassmann","year":"2002","unstructured":"Grassmann, W.K., Usmani, R.A.: Real eigenvalues of certain tridiagonal matrix polynomials, with queueing applications. Linear Algebra Appl. 342(1), 93\u2013106 (2002)","journal-title":"Linear Algebra Appl."},{"key":"1774_CR4","volume-title":"Discret. Math. Its Appl.","author":"KH Rosen","year":"2007","unstructured":"Rosen, K.H.: Discret. Math. Its Appl. McGraw-Hill, New York (2007)"},{"issue":"1","key":"1774_CR5","first-page":"245","volume":"172","author":"J Rimas","year":"2006","unstructured":"Rimas, J.: On computing of arbitrary positive integer powers for one type of symmetric tridiagonal matrices of even order ii. Appl. Math. Comput. 172(1), 245\u2013251 (2006)","journal-title":"Appl. Math. Comput."},{"issue":"1","key":"1774_CR6","first-page":"676","volume":"174","author":"J Rimas","year":"2006","unstructured":"Rimas, J.: On computing of arbitrary positive integer powers for one type of symmetric tridiagonal matrices of odd order ii. Appl. Math. Comput. 174(1), 676\u2013683 (2006)","journal-title":"Appl. Math. Comput."},{"issue":"2","key":"1774_CR7","first-page":"1120","volume":"181","author":"J Rimas","year":"2006","unstructured":"Rimas, J.: On computing of arbitrary positive integer powers for one type of even order skew-symmetric tridiagonal matrices with eigenvalues on immaginary axis ii. Appl. Math. Comput. 181(2), 1120\u20131125 (2006)","journal-title":"Appl. Math. Comput."},{"issue":"22","key":"1774_CR8","doi-asserted-by":"publisher","first-page":"1061","DOI":"10.12988\/imf.2006.06086","volume":"1","author":"DK Salkuyeh","year":"2006","unstructured":"Salkuyeh, D.K.: Positive integer powers of the tridiagonal toeplitz matrices. Int. Math. Forum 1(22), 1061\u20131065 (2006)","journal-title":"Int. Math. Forum"},{"issue":"2","key":"1774_CR9","first-page":"1466","volume":"190","author":"J Rimas","year":"2007","unstructured":"Rimas, J.: On computing of arbitrary positive integer powers for one type of even order tridiagonal matrices with eigenvalues on immaginary axis ii. Appl. Math. Comput. 190(2), 1466\u20131471 (2007)","journal-title":"Appl. Math. Comput."},{"issue":"1","key":"1774_CR10","first-page":"133","volume":"202","author":"J Guti\u00e9rrez-Guti\u00e9rrez","year":"2008","unstructured":"Guti\u00e9rrez-Guti\u00e9rrez, J.: Positive integer powers of certain tridiagonal matrices. Appl. Math. Comput. 202(1), 133\u2013140 (2008)","journal-title":"Appl. Math. Comput."},{"issue":"21","key":"1774_CR11","doi-asserted-by":"publisher","first-page":"10448","DOI":"10.1016\/j.amc.2013.04.030","volume":"219","author":"A \u00d6tele\u015f","year":"2013","unstructured":"\u00d6tele\u015f, A., Akbulak, M.: Positive integer powers of certain complex tridiagonal matrices. Appl. Math. Comput. Appl. Math. Comput. 219(21), 10448\u201310455 (2013)","journal-title":"Appl. Math. Comput. Appl. Math. Comput."},{"issue":"12","key":"1774_CR12","first-page":"4456","volume":"215","author":"MEA El-Mikkawy","year":"2010","unstructured":"El-Mikkawy, M.E.A., Sogabe, T.: A new family of $$k$$-fibonacci numbers. Appl. Math. Comput. 215(12), 4456\u20134461 (2010)","journal-title":"Appl. Math. Comput."},{"issue":"6","key":"1774_CR13","first-page":"2740","volume":"218","author":"T Sogabe","year":"2011","unstructured":"Sogabe, T., El-Mikkawy, M.E.A.: Fast block diagonalization of $$k$$-tridiagonal matrices. Appl. Math. Comput. 218(6), 2740\u20132743 (2011)","journal-title":"Appl. Math. Comput."},{"issue":"1","key":"1774_CR14","doi-asserted-by":"publisher","first-page":"116","DOI":"10.1016\/j.camwa.2012.11.001","volume":"65","author":"JT Jia","year":"2013","unstructured":"Jia, J.T., Sogabe, T., El-Mikkawy, M.E.A.: Inversion of $$k$$-tridiagonal matrices with toeplitz structure. Comput. Math. Appl. 65(1), 116\u2013125 (2013)","journal-title":"Comput. Math. Appl."},{"issue":"2","key":"1774_CR15","doi-asserted-by":"publisher","first-page":"289","DOI":"10.12732\/ijpam.v96i2.10","volume":"96","author":"F Y\u0131lmaz","year":"2014","unstructured":"Y\u0131lmaz, F., Sogabe, T.: A note on symmetric $$k$$-tridiagonal matrix family and the fibonacci numbers. Int. J. Pure and Appl. Math. 96(2), 289\u2013298 (2014)","journal-title":"Int. J. Pure and Appl. Math."},{"key":"1774_CR16","doi-asserted-by":"publisher","first-page":"34","DOI":"10.1016\/j.aml.2014.12.018","volume":"44","author":"MEA El-Mikkawy","year":"2015","unstructured":"El-Mikkawy, M.E.A., Atlan, F.: A new recursive algorithm for inverting general $$k$$-tridiagonal matrices. Appl. Math. Lett. 44, 34\u201339 (2015)","journal-title":"Appl. Math. Lett."},{"key":"1774_CR17","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.jocs.2018.12.009","volume":"31","author":"A Tanasescu","year":"2019","unstructured":"Tanasescu, A., Popescu, P.G.: A fast singular value decomposition algorithm of general $$k$$-tridiagonal matrices. J. Comput. Sci. 31, 1\u20135 (2019)","journal-title":"J. Comput. Sci."},{"issue":"3","key":"1774_CR18","first-page":"298","volume":"57","author":"CM da Fonseca","year":"2020","unstructured":"da Fonseca, C.M., Kowalenko, V., Losonczi, L.: Ninety years of $$k$$-tridiagonal matrices. Stud. Sci. Math. Hung. 57(3), 298\u2013311 (2020)","journal-title":"Stud. Sci. Math. Hung."},{"issue":"1","key":"1774_CR19","first-page":"514","volume":"189","author":"R Witu\u0142a","year":"2007","unstructured":"Witu\u0142a, R., S\u0142ota, D.: On computing the determinants and inverses of some special type of tridiagonal and constant-diagonals matrices. Appl. Math. Comput. 189(1), 514\u2013527 (2007)","journal-title":"Appl. Math. Comput."},{"issue":"1","key":"1774_CR20","first-page":"644","volume":"270","author":"CM da Fonseca","year":"2015","unstructured":"da Fonseca, C.M., Y\u0131lmaz, F.: Some comments on $$k$$-tridiagonal matrices: determinant, spectra, and inversion. Appl. Math. Comput. 270(1), 644\u2013647 (2015)","journal-title":"Appl. Math. Comput."},{"key":"1774_CR21","doi-asserted-by":"publisher","first-page":"112793","DOI":"10.1016\/j.cam.2020.112793","volume":"375","author":"CM da Fonseca","year":"2020","unstructured":"da Fonseca, C.M.: Comments on the spectrum of a tridiagonal $$k$$-toeplitz matrix. J. Comput. Appl. Math. 375, 112793\u2013112795 (2020)","journal-title":"J. Comput. Appl. Math."},{"issue":"2","key":"1774_CR22","doi-asserted-by":"publisher","first-page":"513","DOI":"10.12732\/ijpam.v106i2.14","volume":"106","author":"A Ohashi","year":"2016","unstructured":"Ohashi, A., Sogabe, T., Usuda, T.S.: Fast block diagonalization of $$(k, k^{\\prime })$$-pentadiagonal matrices. Int. J. Pure Appl. Math. 106(2), 513\u2013523 (2016)","journal-title":"Int. J. Pure Appl. Math."},{"key":"1774_CR23","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719512","volume-title":"Matrix Anal. Appl. Linear. Algebra","author":"CD Meyer","year":"2000","unstructured":"Meyer, C.D.: Matrix Anal. Appl. Linear. Algebra. SIAM, Philadelphia (2000)"},{"key":"1774_CR24","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1007\/BF00051649","volume":"60","author":"L Losonczi","year":"1992","unstructured":"Losonczi, L.: Eigenvalues and eigenvectors of some tridiagonal matrices. Acta Math. Hung. 60, 309\u2013322 (1992)","journal-title":"Acta Math. Hung."},{"key":"1774_CR25","doi-asserted-by":"publisher","first-page":"282","DOI":"10.1016\/j.laa.2020.09.014","volume":"608","author":"L Losonczi","year":"2021","unstructured":"Losonczi, L.: Eigenpairs of some imperfect pentadiagonal toeplitz matrices. Linear. Algebra. Its appl. 608, 282\u2013298 (2021)","journal-title":"Linear. Algebra. Its appl."}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-022-01774-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-022-01774-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-022-01774-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,1,27]],"date-time":"2023-01-27T22:43:16Z","timestamp":1674859396000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-022-01774-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,19]]},"references-count":25,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2023,2]]}},"alternative-id":["1774"],"URL":"https:\/\/doi.org\/10.1007\/s12190-022-01774-3","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"value":"1598-5865","type":"print"},{"value":"1865-2085","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,7,19]]},"assertion":[{"value":"14 May 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 May 2022","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 July 2022","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 July 2022","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}