{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,1,10]],"date-time":"2024-01-10T16:41:54Z","timestamp":1704904914626},"reference-count":32,"publisher":"Hindawi Limited","license":[{"start":{"date-parts":[[2021,12,22]],"date-time":"2021-12-22T00:00:00Z","timestamp":1640131200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2021,12,22]]},"abstract":"In this work, the Catalan transformation (CT) of \n \n k<\/mi>\n <\/math>\n <\/jats:inline-formula>-balancing sequences, \n \n \n \n \n \n \n \n B<\/mi>\n <\/mrow>\n \n k<\/mi>\n ,<\/mo>\n n<\/mi>\n <\/mrow>\n <\/msub>\n <\/mrow>\n <\/mfenced>\n <\/mrow>\n \n n<\/mi>\n \u2265<\/mo>\n 0<\/mn>\n <\/mrow>\n <\/msub>\n <\/math>\n <\/jats:inline-formula>, is introduced. Furthermore, the obtained Catalan transformation \n \n \n \n \n \n C<\/mi>\n \n \n B<\/mi>\n <\/mrow>\n \n k<\/mi>\n ,<\/mo>\n n<\/mi>\n <\/mrow>\n <\/msub>\n <\/mrow>\n <\/mfenced>\n <\/mrow>\n \n n<\/mi>\n \u2265<\/mo>\n 0<\/mn>\n <\/mrow>\n <\/msub>\n <\/math>\n <\/jats:inline-formula> was shown as the product of lower triangular matrices called Catalan matrices and the matrix of \n \n k<\/mi>\n <\/math>\n <\/jats:inline-formula>-balancing sequences, \n \n \n \n \n \n \n \n B<\/mi>\n <\/mrow>\n \n k<\/mi>\n ,<\/mo>\n n<\/mi>\n <\/mrow>\n <\/msub>\n <\/mrow>\n <\/mfenced>\n <\/mrow>\n \n n<\/mi>\n \u2265<\/mo>\n 0<\/mn>\n <\/mrow>\n <\/msub>\n <\/math>\n <\/jats:inline-formula>, which is an \n \n n<\/mi>\n \u00d7<\/mo>\n 1<\/mn>\n <\/math>\n <\/jats:inline-formula> matrix. Apart from that, the Hankel transform is applied further to calculate the determinant of the matrices formed from \n \n \n \n \n \n C<\/mi>\n \n \n B<\/mi>\n <\/mrow>\n \n k<\/mi>\n ,<\/mo>\n n<\/mi>\n <\/mrow>\n <\/msub>\n <\/mrow>\n <\/mfenced>\n <\/mrow>\n \n n<\/mi>\n \u2265<\/mo>\n 0<\/mn>\n <\/mrow>\n <\/msub>\n <\/math>\n <\/jats:inline-formula>.<\/jats:p>","DOI":"10.1155\/2021\/9987314","type":"journal-article","created":{"date-parts":[[2021,12,22]],"date-time":"2021-12-22T19:07:48Z","timestamp":1640200068000},"page":"1-6","source":"Crossref","is-referenced-by-count":1,"title":["Catalan Transform of \n k<\/mi>\n <\/math>-Balancing Sequences"],"prefix":"10.1155","volume":"2021","author":[{"given":"Asim","family":"Patra","sequence":"first","affiliation":[{"name":"Department of Mathematics, National Institute of Technology, Rourkela, Odisha, India"}]},{"ORCID":"http:\/\/orcid.org\/0000-0003-2260-0341","authenticated-orcid":true,"given":"Mohammed K. A.","family":"Kaabar","sequence":"additional","affiliation":[{"name":"Gofa Camp, Near Gofa Industrial College and German Adebabay, Nifas Silk-Lafto, Addis Ababa 26649, Ethiopia"},{"name":"Institute of Mathematical Sciences, Faculty of Science, University of Malaya, Kuala Lumpur 50603, Malaysia"}]}],"member":"98","reference":[{"key":"1","first-page":"98","article-title":"On the square roots of triangular numbers","volume":"37","author":"A. Behera","year":"1999","journal-title":"Fibonacci Quarterly"},{"key":"2","unstructured":"RayP. K.Balancing and cobalancing numbers2009Rourkela, IndiaDepartment of Mathematics, National Institute of TechnologyPh.D. thesis"},{"key":"3","first-page":"185","article-title":"Some fascinating properties of balancing numbers","volume":"194","author":"G. K. Panda","year":"2009","journal-title":"Congressus Numerantium"},{"key":"4","first-page":"41","article-title":"Some links of balancing and cobalancing numbers with pell and associated pell numbers","volume":"61","author":"G. K. Panda","year":"2011","journal-title":"Bulletin Institute Mathematics Academia Sinica (New Series)"},{"key":"5","first-page":"A8","article-title":"Some congruences for balancing and lucas-balancing numbers and their applications","volume":"14","author":"P. K. Ray","year":"2014","journal-title":"Integers"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1007\/s40995-020-00849-9"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.15330\/cmp.13.1.259-271"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.1007\/s40590-020-00293-0"},{"key":"9","first-page":"3","article-title":"Positive integer solutions of some diophantine equations involving Lucas-balancing numbers","volume":"581","author":"A. Patra","year":"2020","journal-title":"Fibonacci Quarterly"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.1007\/s12044-021-00615-w"},{"key":"11","volume-title":"Binomial transform","author":"E. W. Weisstein","year":"2004"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1109\/cicn.2013.89"},{"key":"13","doi-asserted-by":"publisher","DOI":"10.1016\/j.compeleceng.2020.106964"},{"key":"14","doi-asserted-by":"publisher","DOI":"10.1002\/ett.3984"},{"key":"15","doi-asserted-by":"publisher","DOI":"10.1109\/radioelek.2011.5936455"},{"key":"16","first-page":"1","article-title":"A catalan transform and related transformations on integer sequences","volume":"84","author":"P. Barry","year":"2005","journal-title":"Journal of Integer Sequence"},{"key":"17","doi-asserted-by":"publisher","DOI":"10.1142\/s1793557122501194"},{"key":"18","doi-asserted-by":"publisher","DOI":"10.7546\/nntdm.2021.27.1.198-207"},{"key":"19","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2021.111473"},{"key":"20","article-title":"Fibonacci number","author":"P. Chandra","year":"2004"},{"key":"21","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4614-8489-9","volume-title":"Pell and Pell-Lucas Numbers with Applications","author":"T. Koshy","year":"2014"},{"key":"22","doi-asserted-by":"publisher","DOI":"10.4134\/ckms.2013.28.4.827"},{"key":"23","first-page":"70","article-title":"Catalan transform of the k-jacobsthal sequence","volume":"82","author":"M. Ta\u015ftan","year":"2020","journal-title":"Electronic Journal of Mathematical Analysis and Applications"},{"key":"24","first-page":"145","article-title":"Catalan transform of the k-lucas numbers","volume":"13","author":"E. \u00d6zkan","year":"2020","journal-title":"Erzincan University Journal of Science and Technology"},{"key":"25","first-page":"148","article-title":"A new sequence realizing lucas numbers, and the lucas bound","volume":"5","author":"E. \u00d6zkan","year":"2017","journal-title":"Electronic Journal of Mathematical Analysis and Applications"},{"key":"26","doi-asserted-by":"publisher","DOI":"10.1007\/s11042-020-10424-4"},{"key":"27","doi-asserted-by":"publisher","DOI":"10.1016\/j.asej.2016.01.014"},{"key":"28","article-title":"Catalan number","author":"R. Stanley","year":"2004"},{"key":"29","first-page":"1","article-title":"Catalan numbers, and hankel transform, and fibonacci numbers","volume":"51","author":"A. Cvetkovi\u0107","year":"2002","journal-title":"Journal of Integer Sequences"},{"key":"30","article-title":"The on-line encyclopedia of integer sequences","author":"N. J. A. Sloane","year":"1964"},{"key":"31","article-title":"The hankel transform and some of its properties","volume":"4","author":"J. W. Layman","year":"2001","journal-title":"Journal of Integer Sequences"},{"key":"32","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1080\/10652460601092303","article-title":"The Hankel transform of the sum of consecutive generalized catalan numbers","volume":"184","author":"P. M. Rajkov\u00edc","year":"2007","journal-title":"Integral Transforms Special Functions"}],"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2021\/9987314.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2021\/9987314.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2021\/9987314.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,12,22]],"date-time":"2021-12-22T19:08:04Z","timestamp":1640200084000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.hindawi.com\/journals\/ijmms\/2021\/9987314\/"}},"subtitle":[],"editor":[{"given":"Sergejs","family":"Solovjovs","sequence":"additional","affiliation":[]}],"short-title":[],"issued":{"date-parts":[[2021,12,22]]},"references-count":32,"alternative-id":["9987314","9987314"],"URL":"https:\/\/doi.org\/10.1155\/2021\/9987314","relation":{},"ISSN":["1687-0425","0161-1712"],"issn-type":[{"value":"1687-0425","type":"electronic"},{"value":"0161-1712","type":"print"}],"subject":[],"published":{"date-parts":[[2021,12,22]]}}}