{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,8]],"date-time":"2026-02-08T16:57:24Z","timestamp":1770569844491,"version":"3.49.0"},"reference-count":25,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2023,11,11]],"date-time":"2023-11-11T00:00:00Z","timestamp":1699660800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Many different number systems have been the topic of research. One of the recently studied number systems is that of hybrid numbers, which are generalizations of other number systems. In this work, we introduce and study the hybrid hyper k-Pell, hybrid hyper k-Pell\u2013Lucas, and hybrid hyper Modified k-Pell numbers. In order to study these new sequences, we established new properties, generating functions, and the Binet formula of the hyper k-Pell, hyper k-Pell\u2013Lucas, and hyper Modified k-Pell sequences. Thus, we present some algebraic properties, recurrence relations, generating functions, the Binet formulas, and some identities for the hybrid hyper k-Pell, hybrid hyper k-Pell\u2013Lucas, and hybrid hyper Modified k-Pell numbers.<\/jats:p>","DOI":"10.3390\/axioms12111047","type":"journal-article","created":{"date-parts":[[2023,11,13]],"date-time":"2023-11-13T02:02:42Z","timestamp":1699840962000},"page":"1047","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["On Hybrid Hyper k-Pell, k-Pell\u2013Lucas, and Modified k-Pell Numbers"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6079-2458","authenticated-orcid":false,"given":"Elen Viviani Pereira","family":"Spreafico","sequence":"first","affiliation":[{"name":"Institute of Mathematics, Federal University of Mato Grosso do Sul, Campo Grande 79074-460, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6917-5093","authenticated-orcid":false,"given":"Paula","family":"Catarino","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Tr\u00e1s-os-Montes e Alto Douro, 5001-801 Vila Real, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5460-4297","authenticated-orcid":false,"given":"Paulo","family":"Vasco","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Tr\u00e1s-os-Montes e Alto Douro, 5001-801 Vila Real, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2023,11,11]]},"reference":[{"key":"ref_1","first-page":"3173","article-title":"On the Pell, Pell\u2013Lucas and Modified Pell Numbers by Matrix Method","volume":"5","author":"Dasdemir","year":"2011","journal-title":"Appl. Math. Sci."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1080\/00150517.1971.12431004","article-title":"Pell identities","volume":"9","author":"Horadam","year":"1971","journal-title":"Fibonacci Q."},{"key":"ref_3","unstructured":"OEIS Foundation Inc. (2023, November 08). The On-Line Encyclopedia of Integer Sequences. Available online: http:\/\/oeis.org."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Koshy, T. (2001). Fibonacci and Lucas Number with Applications, A Wiley-Interscience Publications.","DOI":"10.1002\/9781118033067"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Koshy, T. (2014). Pell and Pell\u2013Lucas Numbers with Applications, Springer.","DOI":"10.1007\/978-1-4614-8489-9"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1016\/j.chaos.2006.10.022","article-title":"The k-Fibonacci sequence and the Pascal 2-triangle","volume":"33","author":"Plaza","year":"2007","journal-title":"Chaos Solitons Fractals"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1877","DOI":"10.12988\/ijma.2013.35131","article-title":"On some identities and generating functions for k-Pell numbers","volume":"7","author":"Catarino","year":"2013","journal-title":"Int. J. Math. Anal."},{"key":"ref_8","first-page":"97","article-title":"A note on k-Pell, k-Pell\u2013Lucas and Modified k-Pell numbers with arithemetic indexes","volume":"89","author":"Catarino","year":"2020","journal-title":"Acta Math. Univ. Comen."},{"key":"ref_9","first-page":"6031","article-title":"Modified k-Pell- Sequence Some Identities and Ordinary Generating Function","volume":"7","author":"Catarino","year":"2013","journal-title":"Appl. Math. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Falc\u00f3n, S., and Plaza, \u00c1. (2022). Iterated Partial Sums of the k-Fibonacci Sequences. Axioms, 11.","DOI":"10.3390\/axioms11100542"},{"key":"ref_11","first-page":"85","article-title":"On The Products of k-Pell Number and k-Pell Lucas number","volume":"13","author":"Panwar","year":"2017","journal-title":"IOSR J. Math. (IOSR-JM)"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1007\/s00009-022-01983-x","article-title":"k-Pell Numbers as Product of Two Repdigits","volume":"19","author":"Rihane","year":"2022","journal-title":"Mediterr. J. Math."},{"key":"ref_13","first-page":"942","article-title":"A symmetric algorithm for hyperharmonic and Fibonacci numbers","volume":"206","author":"Dil","year":"2008","journal-title":"Appl. Math. Comput."},{"key":"ref_14","first-page":"A60","article-title":"On hyper k-Pell, hyper k-Pell\u2013Lucas and hyper Modified k-Pell sequences","volume":"22","author":"Catarino","year":"2022","journal-title":"Integers"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"289","DOI":"10.2307\/2313129","article-title":"Complex Fibonacci Numbers and Fibonacci Quaternions","volume":"70","author":"Horadam","year":"1963","journal-title":"Am. Math. Mon."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1080\/00150517.1965.12431411","article-title":"Gaussian Fibonacci and Lucas Numbers","volume":"3","author":"Jordan","year":"1965","journal-title":"Fibonacci Q."},{"key":"ref_17","unstructured":"Morales, G.C. (2018). Investigation of generalized hybrid Fibonacci numbers and their properties. arXiv."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1007\/s00006-018-0833-3","article-title":"Introduction to Hybrid Numbers","volume":"28","year":"2018","journal-title":"Adv. Appl. Clifford Algebras"},{"key":"ref_19","first-page":"3","article-title":"The Fibonacci hybrid numbers","volume":"11","author":"Wloch","year":"2019","journal-title":"Util. Math."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Y\u0131lmaz, F., and \u00d6zkan, M. (2022). On the Generalized Gaussian Fibonacci Numbers and Horadam Hybrid Numbers: A Unified Approach. Axioms, 11.","DOI":"10.3390\/axioms11060255"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"91","DOI":"10.7151\/dmgaa.1287","article-title":"The horadam hybrid numbers","volume":"38","year":"2018","journal-title":"Discuss. Math. Gen. Algebra Appl."},{"key":"ref_22","first-page":"11","article-title":"On Pell and Pell\u2013Lucas Hybrid Numbers","volume":"58","author":"Wloch","year":"2018","journal-title":"Comment. Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"276","DOI":"10.2478\/amsil-2018-0009","article-title":"On Jacobsthal and Jacobsthal-Lucas Hybrid Numbers","volume":"33","author":"Wloch","year":"2019","journal-title":"Ann. Math. Silesianae"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"154","DOI":"10.7546\/nntdm.2023.29.1.154-170","article-title":"Hybrid hyper-Fibonacci and hyper-Lucas numbers","volume":"29","author":"Yasemin","year":"2023","journal-title":"Notes Number Theory Discret. Math."},{"key":"ref_25","unstructured":"Spreafico, E.V.P., and Catarino, P. (2023). A note on hyper-Leonardo numbers. Notes Number Theory Discret. Math., submitted."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/11\/1047\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:21:32Z","timestamp":1760131292000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/11\/1047"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,11,11]]},"references-count":25,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2023,11]]}},"alternative-id":["axioms12111047"],"URL":"https:\/\/doi.org\/10.3390\/axioms12111047","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,11,11]]}}}