{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T14:39:58Z","timestamp":1740148798524,"version":"3.37.3"},"reference-count":31,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2024,1,8]],"date-time":"2024-01-08T00:00:00Z","timestamp":1704672000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,1,8]],"date-time":"2024-01-08T00:00:00Z","timestamp":1704672000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Nevsehir Haci Bektas Veli University"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2024,2]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Moore\u2013Penrose inverse emerges in statistics, neural networks, machine learning, applied physics, numerical analysis, tensor computations, solving systems of linear equations and in many other disciplines. Especially after the 2000s, the topic of Moore\u2013Penrose inverse has started to attract great attention by researchers and has become a popular subject. In this paper, we investigate the Moore\u2013Penrose inverse of the conditional matrices via convolution product formula. In order to use convolution formula effectively, we derive some useful identities by using some properties of the generalized conditional sequence. Moreover, we express the Moore\u2013Penrose inverse of the conditional matrices in the form of block matrices. Finally, we not only present more general results compared to earlier works, but also provide many novel results using analytical techniques.<\/jats:p>","DOI":"10.1007\/s12190-023-01974-5","type":"journal-article","created":{"date-parts":[[2024,1,8]],"date-time":"2024-01-08T21:22:30Z","timestamp":1704748950000},"page":"417-433","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Inverse and Moore\u2013Penrose inverse of conditional matrices via convolution"],"prefix":"10.1007","volume":"70","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6488-9035","authenticated-orcid":false,"given":"Cahit","family":"K\u00f6me","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yasin","family":"Yazlik","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,1,8]]},"reference":[{"key":"1974_CR1","doi-asserted-by":"publisher","first-page":"365","DOI":"10.1007\/BF02421317","volume":"27","author":"I Fredholm","year":"1903","unstructured":"Fredholm, I.: Sur une classe d\u2019\u00e9quations fonctionnelles. Acta Math. 27, 365\u2013390 (1903)","journal-title":"Acta Math."},{"key":"1974_CR2","first-page":"394","volume":"26","author":"EH Moore","year":"1920","unstructured":"Moore, E.H.: On the reciprocal of the general algebraic matrix. Bull. Am. Math. Soc. 26, 394\u2013395 (1920)","journal-title":"Bull. Am. Math. Soc."},{"key":"1974_CR3","doi-asserted-by":"crossref","unstructured":"Penrose, R.: A generalized inverse for matrices. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 51, pp. 406\u2013413. Cambridge University Press (1955)","DOI":"10.1017\/S0305004100030401"},{"key":"1974_CR4","volume-title":"Generalized Inverses: Theory and Applications","author":"A Ben-Israel","year":"2003","unstructured":"Ben-Israel, A., Greville, T.N.: Generalized Inverses: Theory and Applications. Springer, Berlin (2003)"},{"key":"1974_CR5","doi-asserted-by":"publisher","DOI":"10.1007\/978-981-13-0146-9","volume-title":"Generalized Inverses: Theory and Computations","author":"G Wang","year":"2018","unstructured":"Wang, G., Wei, Y., Qiao, S., Lin, P., Chen, Y.: Generalized Inverses: Theory and Computations. Springer, Berlin (2018)"},{"issue":"1","key":"1974_CR6","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1140\/epjh\/s13129-021-00011-y","volume":"46","author":"OM Baksalary","year":"2021","unstructured":"Baksalary, O.M., Trenkler, G.: The Moore\u2013Penrose inverse: a hundred years on a frontline of physics research. Eur. Phys. J. H 46(1), 1\u201310 (2021)","journal-title":"Eur. Phys. J. H"},{"key":"1974_CR7","first-page":"1","volume":"71","author":"O Maria Baksalary","year":"2022","unstructured":"Maria Baksalary, O., Sivakumar, K., Trenkler, G.: On the Moore\u2013Penrose inverse of a sum of matrices. Linear and Multilinear Algebra 71, 1\u201317 (2022)","journal-title":"Linear and Multilinear Algebra"},{"key":"1974_CR8","unstructured":"Courrieu, P.: Fast computation of Moore\u2013Penrose inverse matrices. Neural Inf. Process. Lett. Rev. 8(2) (2005)"},{"issue":"1","key":"1974_CR9","doi-asserted-by":"publisher","first-page":"15","DOI":"10.1137\/1002004","volume":"2","author":"T Greville","year":"1960","unstructured":"Greville, T.: Some applications of the pseudoinverse of a matrix. SIAM Rev. 2(1), 15\u201322 (1960)","journal-title":"SIAM Rev."},{"issue":"1","key":"1974_CR10","doi-asserted-by":"publisher","first-page":"33","DOI":"10.4134\/JKMS.2011.48.1.033","volume":"48","author":"M Miladinovic","year":"2011","unstructured":"Miladinovic, M., Stanimirovic, P.: Singular case of generalized Fibonacci and Lucas matrices. J. Korean Math. Soc. 48(1), 33\u201348 (2011)","journal-title":"J. Korean Math. Soc."},{"issue":"1","key":"1974_CR11","doi-asserted-by":"publisher","first-page":"29","DOI":"10.36045\/bbms\/1590199301","volume":"27","author":"B Radi\u010di\u0107","year":"2020","unstructured":"Radi\u010di\u0107, B.: The inverse and the Moore\u2013Penrose inverse of a $$k$$-circulant matrix with binomial coefficients. Bull. Belg. Math. Soc. Simon Stevin 27(1), 29\u201342 (2020)","journal-title":"Bull. Belg. Math. Soc. Simon Stevin"},{"issue":"2","key":"1974_CR12","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1007\/s00607-010-0133-9","volume":"92","author":"Y Zhang","year":"2011","unstructured":"Zhang, Y., Yang, Y., Tan, N., Cai, B.: Zhang neural network solving for time-varying full-rank matrix Moore\u2013Penrose inverse. Computing 92(2), 97\u2013121 (2011)","journal-title":"Computing"},{"issue":"4","key":"1974_CR13","doi-asserted-by":"publisher","first-page":"686","DOI":"10.1080\/03081087.2015.1083933","volume":"64","author":"L Sun","year":"2016","unstructured":"Sun, L., Zheng, B., Bu, C., Wei, Y.: Moore\u2013Penrose inverse of tensors via Einstein product. Linear Multilinear Algebra 64(4), 686\u2013698 (2016)","journal-title":"Linear Multilinear Algebra"},{"issue":"3","key":"1974_CR14","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s40314-019-0893-6","volume":"38","author":"H Ma","year":"2019","unstructured":"Ma, H., Li, N., Stanimirovi\u0107, P.S., Katsikis, V.N.: Perturbation theory for Moore\u2013Penrose inverse of tensor via Einstein product. Comput. Appl. Math. 38(3), 1\u201324 (2019)","journal-title":"Comput. Appl. Math."},{"key":"1974_CR15","doi-asserted-by":"publisher","DOI":"10.1002\/9781118033067","volume-title":"Fibonacci and Lucas Numbers with Applications","author":"T Koshy","year":"2001","unstructured":"Koshy, T.: Fibonacci and Lucas Numbers with Applications. Wiley, Hoboken (2001)"},{"issue":"1","key":"1974_CR16","doi-asserted-by":"publisher","first-page":"38","DOI":"10.1016\/j.chaos.2006.10.022","volume":"33","author":"S Falcon","year":"2007","unstructured":"Falcon, S., Plaza, \u00c1.: The $$k$$-Fibonacci sequence and the Pascal $$2$$-triangle. Chaos Solitons Fractals 33(1), 38\u201349 (2007)","journal-title":"Chaos Solitons Fractals"},{"issue":"3","key":"1974_CR17","first-page":"361","volume":"46","author":"P Catarino","year":"2017","unstructured":"Catarino, P., Campos, H.: Incomplete $$k$$-Pell, $$k$$-Pell\u2013Lucas and modified $$k$$-Pell numbers. Hacet. J. Math. Stat. 46(3), 361\u2013372 (2017)","journal-title":"Hacet. J. Math. Stat."},{"issue":"1","key":"1974_CR18","doi-asserted-by":"publisher","first-page":"36","DOI":"10.1016\/j.camwa.2011.10.055","volume":"63","author":"Y Yazlik","year":"2012","unstructured":"Yazlik, Y., Taskara, N.: A note on generalized $$k$$-Horadam sequence. Comput. Math. Appl. 63(1), 36\u201341 (2012)","journal-title":"Comput. Math. Appl."},{"issue":"6","key":"1974_CR19","doi-asserted-by":"publisher","first-page":"639","DOI":"10.1515\/INTEG.2009.051","volume":"9","author":"M Edson","year":"2009","unstructured":"Edson, M., Yayenie, O.: A new generalization of Fibonacci sequence and extended Binet\u2019s formula. Integers 9(6), 639\u2013654 (2009)","journal-title":"Integers"},{"issue":"4","key":"1974_CR20","first-page":"657","volume":"25","author":"Y Yazlik","year":"2018","unstructured":"Yazlik, Y., K\u00f6me, C., Madhusudanan, V.: A new generalization of Fibonacci and Lucas $$p$$-numbers. J. Comput. Anal. Appl. 25(4), 657\u2013669 (2018)","journal-title":"J. Comput. Anal. Appl."},{"key":"1974_CR21","first-page":"526","volume":"245","author":"G Bilgici","year":"2014","unstructured":"Bilgici, G.: Two generalizations of Lucas sequence. Appl. Math. Comput. 245, 526\u2013538 (2014)","journal-title":"Appl. Math. Comput."},{"key":"1974_CR22","first-page":"1","volume":"49","author":"E Tan","year":"2019","unstructured":"Tan, E., Leung, H.-H.: A note on congruence properties of the generalized bi-periodic Horadam sequence. Hacet. J. Math. Stat. 49, 1\u201310 (2019)","journal-title":"Hacet. J. Math. Stat."},{"issue":"12","key":"1974_CR23","first-page":"5603","volume":"217","author":"O Yayenie","year":"2011","unstructured":"Yayenie, O.: A note on generalized Fibonacci sequences. Appl. Math. Comput. 217(12), 5603\u20135611 (2011)","journal-title":"Appl. Math. Comput."},{"issue":"6","key":"1974_CR24","first-page":"739","volume":"11","author":"M Edson","year":"2011","unstructured":"Edson, M., Lewis, S., Yayenie, O.: The $$k$$-periodic Fibonacci sequence and an extended Binet\u2019s formula. J. Integers 11(6), 739\u2013751 (2011)","journal-title":"J. Integers"},{"issue":"1","key":"1974_CR25","doi-asserted-by":"publisher","first-page":"392","DOI":"10.1186\/s13662-019-2327-6","volume":"2019","author":"D Marques","year":"2019","unstructured":"Marques, D., Trojovsk\u1ef3, P.: On characteristic polynomial of higher order generalized Jacobsthal numbers. Adv. Differ. Equ. 2019(1), 392 (2019)","journal-title":"Adv. Differ. Equ."},{"issue":"3","key":"1974_CR26","first-page":"1","volume":"51","author":"NR Ait-Amrane","year":"2022","unstructured":"Ait-Amrane, N.R., Belbachir, H.: Bi-periodic $$r$$-Fibonacci sequence and bi-periodic $$r$$-Lucas sequence of type $$s$$. Hacet. J. Math. Stat. 51(3), 1\u201320 (2022)","journal-title":"Hacet. J. Math. Stat."},{"issue":"1","key":"1974_CR27","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1186\/s13662-020-2507-4","volume":"2020","author":"E Tan","year":"2020","unstructured":"Tan, E., Leung, H.-H.: Some basic properties of the generalized bi-periodic Fibonacci and Lucas sequences. Adv. Differ. Equ. 2020(1), 1\u201311 (2020)","journal-title":"Adv. Differ. Equ."},{"issue":"10","key":"1974_CR28","doi-asserted-by":"publisher","first-page":"1756","DOI":"10.1080\/00207160.2015.1074189","volume":"93","author":"S-Q Shen","year":"2016","unstructured":"Shen, S.-Q., He, J.-J.: Moore\u2013Penrose inverse of generalized Fibonacci matrix and its applications. Int. J. Comput. Math. 93(10), 1756\u20131770 (2016)","journal-title":"Int. J. Comput. Math."},{"issue":"4","key":"1974_CR29","doi-asserted-by":"publisher","first-page":"929","DOI":"10.7153\/oam-2017-11-65","volume":"11","author":"S Shen","year":"2017","unstructured":"Shen, S., Liu, W., Feng, L.: Inverse and Moore\u2013Penrose inverse of Toeplitz matrices with classical Horadam numbers. Oper. Matrices 11(4), 929\u2013939 (2017)","journal-title":"Oper. Matrices"},{"key":"1974_CR30","doi-asserted-by":"publisher","DOI":"10.1007\/s13226-022-00352-4","author":"C K\u00f6me","year":"2022","unstructured":"K\u00f6me, C.: Moore\u2013Penrose inverse of the singular conditional matrices and its applications. Indian J. Pure Appl. Math. (2022). https:\/\/doi.org\/10.1007\/s13226-022-00352-4","journal-title":"Indian J. Pure Appl. Math."},{"issue":"7","key":"1974_CR31","doi-asserted-by":"publisher","first-page":"1519","DOI":"10.1080\/00207160.2010.521546","volume":"88","author":"P Stanimirovi\u0107","year":"2011","unstructured":"Stanimirovi\u0107, P., Miladinovi\u0107, M.: Inversion of the generalized Fibonacci matrix by convolution. Int. J. Comput. Math. 88(7), 1519\u20131532 (2011)","journal-title":"Int. J. Comput. Math."}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-023-01974-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-023-01974-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-023-01974-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,6]],"date-time":"2024-02-06T13:23:16Z","timestamp":1707225796000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-023-01974-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,1,8]]},"references-count":31,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2024,2]]}},"alternative-id":["1974"],"URL":"https:\/\/doi.org\/10.1007\/s12190-023-01974-5","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"type":"print","value":"1598-5865"},{"type":"electronic","value":"1865-2085"}],"subject":[],"published":{"date-parts":[[2024,1,8]]},"assertion":[{"value":"18 October 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 December 2023","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 December 2023","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 January 2024","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"}}]}}