{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,14]],"date-time":"2026-01-14T16:01:54Z","timestamp":1768406514388,"version":"3.49.0"},"reference-count":7,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2023,11,24]],"date-time":"2023-11-24T00:00:00Z","timestamp":1700784000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Fuzhou University","award":["0490-50011045"],"award-info":[{"award-number":["0490-50011045"]}]},{"name":"Fuzhou University","award":["JAT1900018"],"award-info":[{"award-number":["JAT1900018"]}]},{"name":"Educational Commission of Fujian Province","award":["0490-50011045"],"award-info":[{"award-number":["0490-50011045"]}]},{"name":"Educational Commission of Fujian Province","award":["JAT1900018"],"award-info":[{"award-number":["JAT1900018"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We define the incomplete generalized bivariate Fibonacci p-polynomials and the incomplete generalized bivariate Lucas p-polynomials. We study their recursive relations and derive an interesting relationship through their generating functions. Subsequently, we prove an incomplete version of the well-known Fibonacci\u2013Lucas relation and make some extensions to the relation involving incomplete generalized bivariate Fibonacci and Lucas p-polynomials. An argument about going from the regular to the incomplete Fibonacci\u2013Lucas relation is discussed. We provide a relation involving the incomplete Leonardo and the incomplete Lucas\u2013Leonardo p-numbers as an illustration.<\/jats:p>","DOI":"10.3390\/sym15122113","type":"journal-article","created":{"date-parts":[[2023,11,24]],"date-time":"2023-11-24T11:19:46Z","timestamp":1700824786000},"page":"2113","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A Note on Incomplete Fibonacci\u2013Lucas Relations"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0009-0007-1298-9433","authenticated-orcid":false,"given":"Jingyang","family":"Zhong","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Fuzhou University, Fuzhou 350108, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jialing","family":"Yao","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Fuzhou University, Fuzhou 350108, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7548-3607","authenticated-orcid":false,"given":"Chan-Liang","family":"Chung","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Fuzhou University, Fuzhou 350108, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,11,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1007\/BF02845088","article-title":"Incomplete Fibonacci and Lucas numbers","volume":"45","author":"Filipponi","year":"1996","journal-title":"Rend. Del Circ. Mat. Palermo"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"236","DOI":"10.4169\/amer.math.monthly.121.03.236","article-title":"A polynomial parent to a Fibonacci-Lucas relation","volume":"3","author":"Sury","year":"2014","journal-title":"Am. Math. Mon."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Chung, C.-L. (2019). Some polynomial sequence relations. Mathematics, 7.","DOI":"10.3390\/math7080750"},{"key":"ref_4","first-page":"A38","article-title":"Extensions of Sury\u2019s relation involving Fibonacci k-step and Lucas k-step polynomials","volume":"23","author":"Chung","year":"2023","journal-title":"Integers"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"840345","DOI":"10.1155\/2012\/840345","article-title":"Incomplete bivariate Fibonacci and Lucas p-polynomials","volume":"2012","author":"Tasci","year":"2012","journal-title":"Discret. Dyn. Nat. Soc."},{"key":"ref_6","first-page":"363","article-title":"Incomplete generalized Fibonacci and Lucas polynomials","volume":"44","year":"2015","journal-title":"Hacet. J. Math. Stat."},{"key":"ref_7","first-page":"A7","article-title":"On Leonardo p-numbers","volume":"23","author":"Tan","year":"2023","journal-title":"Integers"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/12\/2113\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:29:49Z","timestamp":1760131789000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/12\/2113"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,11,24]]},"references-count":7,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2023,12]]}},"alternative-id":["sym15122113"],"URL":"https:\/\/doi.org\/10.3390\/sym15122113","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,11,24]]}}}