{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T16:29:45Z","timestamp":1777566585137,"version":"3.51.4"},"reference-count":55,"publisher":"Hacettepe University","issue":"4","funder":[{"name":"University of Beira Interior, Portugal; University of Tr\u00e1s-os-Montes e Alto Douro, Portugal","award":["UIDB\/00212\/2020, MTM2017-83506-C2-2-P, UIDB\/00013\/2020, UIDP\/00013\/2020, UID\/CED\/00194\/2020"],"award-info":[{"award-number":["UIDB\/00212\/2020, MTM2017-83506-C2-2-P, UIDB\/00013\/2020, UIDP\/00013\/2020, UID\/CED\/00194\/2020"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"accepted":{"date-parts":[[2023,9,3]]},"abstract":"<jats:p xml:lang=\"en\">In the present work, a new sequence of quaternions related to the Leonardo numbers -- named the Leonardo quaternions sequence -- is defined and studied. Binet's formula and certain sum and binomial-sum identities, some of which derived from the mentioned formula, are established. Tagiuri-Vajda's identity and, as consequences, Catalan's identity, d'Ocagne's identity and Cassini's identity are presented. Furthermore, applying Catalan's identity, and the connection between composition algebras and vector cross product algebras, Gelin-Ces\u00e0ro's identity is also stated and proved.  Finally, the generating function, the exponential generating function and the Poisson generating function are deduced.  In addition to the results on Leonardo quaternions, known results on Leonardo numbers and on Fibonacci quaternions are extended.<\/jats:p>","DOI":"10.15672\/hujms.1197693","type":"journal-article","created":{"date-parts":[[2024,8,13]],"date-time":"2024-08-13T05:24:46Z","timestamp":1723526686000},"page":"1001-1023","source":"Crossref","is-referenced-by-count":9,"title":["On the Leonardo quaternions sequence"],"prefix":"10.15672","volume":"53","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0266-7055","authenticated-orcid":true,"given":"Patr\u00edcia","family":"Beites","sequence":"first","affiliation":[{"name":"University of Beira Interior"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6917-5093","authenticated-orcid":true,"given":"Paula Maria Machado Cruz","family":"Catarino","sequence":"additional","affiliation":[{"name":"University of Tr\u00e1s-os-Montes e Alto Douro"}]}],"member":"6337","published-online":{"date-parts":[[2024,8,27]]},"reference":[{"key":"ref1","unstructured":"[1] B. Aloui and A. Boussayoud, Generating functions of the product of the k-Fibonacci\r\nand k-Pell numbers and Chebyshev polynomials of the third and fourth kind, Math.\r\nEng. Sci. Aerosp. 12, 245-257, 2021."},{"key":"ref2","doi-asserted-by":"crossref","unstructured":"[2] F. Alves and R. Vieira, The Newton fractal\u2019s Leonardo sequence study with the Google\r\nColab, Int. Electron. J. Math. Educ. 15, em0575, 2020.","DOI":"10.29333\/iejme\/6440"},{"key":"ref3","doi-asserted-by":"crossref","unstructured":"[3] M. Akyi\u011fit, H.H. K\u00f6sal and M. Tosun, Split Fibonacci quaternions, Adv. Appl. Clifford\r\nAlgebras 23, 535-545, 2013.","DOI":"10.1007\/s00006-013-0401-9"},{"key":"ref4","doi-asserted-by":"crossref","unstructured":"[4] M. Akyi\u011fit, H.H. K\u00f6sal and M. Tosun, Fibonacci generalized quaternions, Adv. Appl.\r\nClifford Algebras 24, 631-641, 2014.","DOI":"10.1007\/s00006-014-0458-0"},{"key":"ref5","unstructured":"[5] Y. Alp and E.G. Ko\u00e7er, Some properties of Leonardo numbers, Konuralp J. Math. 9,\r\n183-189, 2021."},{"key":"ref6","doi-asserted-by":"crossref","unstructured":"[6] P.D. Beites and A.P. Nicol\u00e1s, An associative triple system of the second kind, Commun.\r\nAlgebra 44, 5027-5043, 2016.","DOI":"10.1080\/00927872.2016.1149185"},{"key":"ref7","doi-asserted-by":"crossref","unstructured":"[7] P.D. Beites and A.P. Nicol\u00e1s, A note on standard composition algebras of types II and\r\nIII, Adv. Appl. Clifford Algebras 27, 955-964, 2017.","DOI":"10.1007\/s00006-016-0668-8"},{"key":"ref8","doi-asserted-by":"crossref","unstructured":"[8] P.D. Beites, A.P. Nicol\u00e1s, P. Saraiva and J. Vit\u00f3ria, Vector cross product differential\r\nand difference equations in $\\mathbb{R}^3$and in$\\mathbb{R}^7$, Electron. J. Linear Algebra 34, 675-686,\r\n2018.","DOI":"10.13001\/1081-3810.3843"},{"key":"ref9","doi-asserted-by":"crossref","unstructured":"[9] G. Bilgici and P. Catarino, Unrestricted pell and pell-lucas quaternions, Int. J. Math.\r\nSyst. Sci. 1, Article 816, 2018.","DOI":"10.24294\/ijmss.v1i3.816"},{"key":"ref10","unstructured":"[10] G. Bilgici, U. Toke\u015fer and Z. \u00dcnal, k-Fibonacci and k-Lucas generalized quaternions,\r\nKonuralp J. Math. 5, 102-113, 2017."},{"key":"ref11","doi-asserted-by":"crossref","unstructured":"[11] T.R. Blackman and S. Lemurell, Spectral correspondences for Maass waveforms on\r\nquaternion groups, J. Number Theory 158, 1-22, 2016.","DOI":"10.1016\/j.jnt.2015.05.017"},{"key":"ref12","doi-asserted-by":"crossref","unstructured":"[12] I. Ca\u00e7\u00e3o, H.R. Malonek and G. Tomaz, Shifted Generalized Pascal Matrices in the\r\nContext of Clifford Algebra-Valued Polynomial Sequences, in: Computational Science\r\nand its Applications, Lecture Notes in Computer Science 10405, Springer, 2017.","DOI":"10.1007\/978-3-319-62395-5_28"},{"key":"ref13","doi-asserted-by":"crossref","unstructured":"[13] N.D. Cahill, J.R. D\u2019Errico, D.A. Narayan and J.Y. Narayan, Fibonacci determinants,\r\nColl. Math. J. 33, 221-225, 2002.","DOI":"10.1080\/07468342.2002.11921945"},{"key":"ref14","doi-asserted-by":"crossref","unstructured":"[14] P. Catarino, The modified Pell and the modified k-Pell quaternions and octonions,\r\nAdv. Appl. Clifford Algebras 26, 577-590, 2016.","DOI":"10.1007\/s00006-015-0611-4"},{"key":"ref15","unstructured":"[15] P. Catarino and A. Borges, On Leonardo numbers, Acta Math. Univ. Comen. 89,\r\n75-86, 2020."},{"key":"ref16","unstructured":"[16] P. Catarino and A. Borges, A note on incomplete Leonardo numbers, Integers 20,\r\narticle A43, 2020."},{"key":"ref17","doi-asserted-by":"crossref","unstructured":"[17] P. Catarino and H. Campos, Incomplete k-Pell, k-Pell-Lucas and modified k-Pell\r\nnumbers, Hacet. J. Math. Stat. 46, 361-372, 2017.","DOI":"10.15672\/HJMS.20164518616"},{"key":"ref18","doi-asserted-by":"crossref","unstructured":"[18] P. Catarino and R. de Almeida, On a quaternionic sequence with Vietoris\u2019 numbers,\r\nFilomat 35, 1065-1086, 2021.","DOI":"10.2298\/FIL2104065C"},{"key":"ref19","doi-asserted-by":"crossref","unstructured":"[19] P. Catarino and R. de Almeida, A note on Vietoris\u2019 number sequence, Mediterr. J.\r\nMath. 19, 1-19, 2022.","DOI":"10.1007\/s00009-021-01952-w"},{"key":"ref20","doi-asserted-by":"crossref","unstructured":"[20] Y. Choo, A generalized quaternion with generalized Fibonacci number components,\r\nAppl. Math. Sci. 14, 31-38, 2020.","DOI":"10.12988\/ams.2020.912160"},{"key":"ref21","doi-asserted-by":"crossref","unstructured":"[21] C.B. \u00c7imen and A. \u0130pek, On Pell quaternions and Pell-Lucas quaternions, Adv. Appl.\r\nClifford Algebras 26, 39-51, 2016.","DOI":"10.1007\/s00006-015-0571-8"},{"key":"ref22","doi-asserted-by":"crossref","unstructured":"[22] J.H. Conway and D.A. Smith, On Quaternions and Octonions: their Geometry, Arithmetic\r\nand Symmetry, A K Peters\/CRC Press, New York, 2003.","DOI":"10.1201\/9781439864180"},{"key":"ref23","doi-asserted-by":"crossref","unstructured":"[23] N. Correia and R. Pacheco, Harmonic maps of finite uniton number and their canonical\r\nelements, Ann. Glob. Anal. Geom. 47, 335-358, 2015.","DOI":"10.1007\/s10455-014-9448-7"},{"key":"ref24","doi-asserted-by":"crossref","unstructured":"[24] A. Da\u015fdemir, Gelin-Ces\u00e0ro identities for Fibonacci and Lucas quaternions, Ann.\r\nUniv. Paedagog. Crac. Stud. Math. XVIII, 137-144, 2019.","DOI":"10.2478\/aupcsm-2019-0010"},{"key":"ref25","doi-asserted-by":"crossref","unstructured":"[25] E.W. Dijkstra, Smoothsort, an alternative for sorting in situ, Sci. Comput. Program.\r\n1, 223-233, 1982.","DOI":"10.1016\/0167-6423(82)90016-8"},{"key":"ref26","unstructured":"[26] G.B. Djordjevi\u0107 and H.M. Srivastava, Some generalizations of certain sequences associated\r\nwith the Fibonacci numbers, J. Indones. Math. Soc. 12, 99-112, 2006."},{"key":"ref27","doi-asserted-by":"crossref","unstructured":"[27] M.I. Falc\u00e3o, F. Miranda, R. Severino and M.J. Soares, Evaluation schemes in the ring\r\nof quaternionic polynomials, BIT Numer. Math. 58, 51-72, 2018.","DOI":"10.1007\/s10543-017-0667-8"},{"key":"ref28","doi-asserted-by":"crossref","unstructured":"[28] N. Jacobson, Composition algebras and their automorphisms, Rend. Circ. Mat.\r\nPalermo 7, 55-80, 1958.","DOI":"10.1007\/BF02854388"},{"key":"ref29","doi-asserted-by":"crossref","unstructured":"[29] S. Halici, On Fibonacci quaternions, Adv. Appl. Clifford Algebras 22, 321-327, 2012.","DOI":"10.1007\/s00006-011-0317-1"},{"key":"ref30","doi-asserted-by":"crossref","unstructured":"[30] S. Halici and G. Cerda-Morales, On Quaternion-Gaussian Fibonacci numbers and\r\ntheir properties, An. St. Univ. Ovidius Constanta, Ser. Mat. 29, 71-82, 2021.","DOI":"10.2478\/auom-2021-0005"},{"key":"ref31","doi-asserted-by":"crossref","unstructured":"[31] S. Halici and A. Karata\u015f, On a generalization for Fibonacci quaternions, Chaos Soliton\r\nFract. 98, 178-182, 2017.","DOI":"10.1016\/j.chaos.2017.03.037"},{"key":"ref32","doi-asserted-by":"crossref","unstructured":"[32] A.F. Horadam, Complex Fibonacci numbers and Fibonacci quaternions, Am. Math.\r\nMon. 70, 289-291, 1963.","DOI":"10.2307\/2313129"},{"key":"ref33","doi-asserted-by":"crossref","unstructured":"[33] A. \u0130pek, On (p, q)-Fibonacci quaternions and their Binet formulas, generating functions\r\nand certain binomial sums, Adv. Appl. Clifford Algebrass 27, 1343-1351, 2017.","DOI":"10.1007\/s00006-016-0704-8"},{"key":"ref34","unstructured":"[34] E. Kilic, D. Tasci and P. Haukkanen, On the generalized Lucas sequences by Hessenberg\r\nmatrices, Ars Combinatoria 95, 383-395, 2010."},{"key":"ref35","unstructured":"[35] D. Knuth, The art of computer programming, Addison Wesley Longman, 1997."},{"key":"ref36","doi-asserted-by":"crossref","unstructured":"[36] T. Koshy, Fibonacci and Lucas numbers with applications, Wiley, 2018.","DOI":"10.1002\/9781118742297"},{"key":"ref37","doi-asserted-by":"crossref","unstructured":"[37] F.S. Leite, The geometry of hypercomplex matrices, Linear Multilinear Algebra 34,\r\n123-132, 1993.","DOI":"10.1080\/03081089308818216"},{"key":"ref38","doi-asserted-by":"crossref","unstructured":"[38] J. Morais and I. Ca\u00e7\u00e3o, Quaternion Zernike spherical polynomials, Math. Comput.\r\n84, 1317-1337, 2015.","DOI":"10.1090\/S0025-5718-2014-02888-3"},{"key":"ref39","unstructured":"[39] B.K. Patel and P.K. Ray, On the properties of $(p, q)$-Fibonacci and $(p, q)$-Lucas quaternions,\r\nMathematical Reports 21, 15-25, 2019."},{"key":"ref40","doi-asserted-by":"crossref","unstructured":"[40] E. Polatl\u0131 and S. Kesim, A note on Catalan\u2019s identity for the k-Fibonacci quaternions,\r\nJ. Integer Seq. 18, article 15.8.2, 2015.","DOI":"10.1186\/s13662-015-0511-x"},{"key":"ref41","doi-asserted-by":"crossref","unstructured":"[41] E. Polatl\u0131, C. Kizilates and S. Kesim, On split k-Fibonacci and k-Lucas quaternions,\r\nAdv. Appl. Clifford Algebras 26, 353-362, 2016.","DOI":"10.1007\/s00006-015-0591-4"},{"key":"ref42","doi-asserted-by":"crossref","unstructured":"[42] E. Polatl\u0131 and Y. Soykan, On Generalized Third-order Jacobsthal numbers, Asian Res.\r\nJ. Math. 17, 1-19, 2021.","DOI":"10.9734\/arjom\/2021\/v17i230270"},{"key":"ref43","doi-asserted-by":"crossref","unstructured":"[43] J.L. Ram\u00edrez, Some combinatorial properties of the k-Fibonacci and the k-Lucas\r\nquaternions, An. St. Univ. Ovidius Constanta, Ser. Mat. 23, 201-212, 2015.","DOI":"10.1515\/auom-2015-0037"},{"key":"ref44","doi-asserted-by":"crossref","unstructured":"[44] N. Saba, A. Boussayoud and K. V. Kanuri, Mersenne Lucas numbers and complete\r\nhomogeneous symmetric functions, J. Math. Comput. Sci. 24, 127-139, 2022.","DOI":"10.22436\/jmcs.024.02.04"},{"key":"ref45","doi-asserted-by":"crossref","unstructured":"[45] R. Ser\u00f4dio, P.D. Beites and J. Vit\u00f3ria, Intersection of a double cone and a line in the\r\nsplit-quaternions context, Adv. Appl. Clifford Algebras 27, 2795-2803, 2017.","DOI":"10.1007\/s00006-017-0796-9"},{"key":"ref46","unstructured":"[46] N.J.A. Sloane, The On-Line Encyclopedia of Integer Sequences, The OEIS Foundation,\r\nhttps:\/\/oeis.org, 2021."},{"key":"ref47","doi-asserted-by":"crossref","unstructured":"[47] A. Szynal-Liana and I. W\u0142och, The Pell quaternions and the Pell octonions, Adv.\r\nAppl. Clifford Algebras 26, 435-440, 2016.","DOI":"10.1007\/s00006-015-0570-9"},{"key":"ref48","doi-asserted-by":"crossref","unstructured":"[48] A. Szynal-Liana and I. W\u0142och, A note on Jacobsthal quaternions, Adv. Appl. Clifford\r\nAlgebras 26, 441-447, 2016.","DOI":"10.1007\/s00006-015-0622-1"},{"key":"ref49","doi-asserted-by":"crossref","unstructured":"[49] E. Tan and H.-H. Leung, Some results on Horadam quaternions, Chaos Soliton Fract.\r\n138, article 109961, 2020.","DOI":"10.1016\/j.chaos.2020.109961"},{"key":"ref50","unstructured":"[50] D. Tasci, On k-Jacobsthal and k-Jacobsthal-Lucas quaternions, Journal of Science and\r\nArts 3, 469-476, 2017."},{"key":"ref51","doi-asserted-by":"crossref","unstructured":"[51] U. Toke\u015fer, Z. \u00dcnal and G. Bilgici, Split Pell and Pell-Lucas quaternions, Adv. Appl.\r\nClifford Algebras 27, 1881-1893, 2017.","DOI":"10.1007\/s00006-016-0747-x"},{"key":"ref52","unstructured":"[52] S. Vajda, Fibonacci & Lucas numbers, and the golden section, Ellis Horwood Ltd.,\r\nChichester, England, 1989."},{"key":"ref53","doi-asserted-by":"crossref","unstructured":"[53] R. Vieira, F. Alves and P. Catarino, Rela\u00e7\u00f5es bidimensionais e identidades da sequ\u00eancia\r\nde Leonardo, Revista Sergipana de Matem\u00e1tica e Educa\u00e7\u00e3o Matem\u00e1tica 4,\r\n156-173, 2019.","DOI":"10.34179\/revisem.v4i2.11863"},{"key":"ref54","doi-asserted-by":"crossref","unstructured":"[54] R. Vieira, M. Mangueira, F. Alves and P. Catarino, A forma matricial dos n\u00fameros\r\nde Leonardo, Ci\u00eancia e Natura 42, article e100, 2020.","DOI":"10.5902\/2179460X41839"},{"key":"ref55","doi-asserted-by":"crossref","unstructured":"[55] T. Ya\u011fmur, Split Jacobsthal and Jacobsthal-Lucas quaternions, Commun. Math. Appl.\r\n10, 429-438, 2019.","DOI":"10.26713\/cma.v10i3.902"}],"container-title":["Hacettepe Journal of Mathematics and Statistics"],"original-title":[],"deposited":{"date-parts":[[2024,9,1]],"date-time":"2024-09-01T21:43:50Z","timestamp":1725227030000},"score":1,"resource":{"primary":{"URL":"http:\/\/dergipark.org.tr\/en\/doi\/10.15672\/hujms.1197693"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,27]]},"references-count":55,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2024,8,27]]}},"URL":"https:\/\/doi.org\/10.15672\/hujms.1197693","relation":{},"ISSN":["2651-477X"],"issn-type":[{"value":"2651-477X","type":"print"}],"subject":[],"published":{"date-parts":[[2024,8,27]]}}}