{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,3]],"date-time":"2026-07-03T13:52:56Z","timestamp":1783086776896,"version":"3.54.6"},"reference-count":44,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2023,12,12]],"date-time":"2023-12-12T00:00:00Z","timestamp":1702339200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,12,12]],"date-time":"2023-12-12T00:00:00Z","timestamp":1702339200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"name":"Scientific Research Program Funded by Shaanxi Provincial Education Department","award":["23JK0661"],"award-info":[{"award-number":["23JK0661"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2024,2]]},"DOI":"10.1007\/s40314-023-02524-2","type":"journal-article","created":{"date-parts":[[2023,12,12]],"date-time":"2023-12-12T17:02:21Z","timestamp":1702400541000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["A new generalization of the Frank matrix and its some properties"],"prefix":"10.1007","volume":"43","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2145-8929","authenticated-orcid":false,"given":"Baijuan","family":"Shi","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Can","family":"K\u0131z\u0131late\u015f","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2023,12,12]]},"reference":[{"issue":"5","key":"2524_CR1","doi-asserted-by":"publisher","first-page":"539","DOI":"10.3390\/math9050539","volume":"9","author":"SY Akbiyik","year":"2021","unstructured":"Akbiyik SY, Akbiyik M, Yilmaz F (2021) One Type of Symmetric Matrix with Harmonic Pell Entries, Its Inversion, Permanents and Some Norms. Mathematics. 9(5):539","journal-title":"Mathematics."},{"key":"2524_CR2","doi-asserted-by":"publisher","first-page":"100","DOI":"10.1016\/j.laa.2021.01.016","volume":"617","author":"E Andrade","year":"2021","unstructured":"Andrade E, Carrasco-Olivera D, Manzaneda C (2021) On circulant like matrices properties involving Horadam, Fibonacci, Jacobsthal and Pell numbers. Linear Algebra Appl 617:100\u2013120","journal-title":"Linear Algebra Appl"},{"key":"2524_CR3","doi-asserted-by":"crossref","unstructured":"Bertaccini D, Ng MK (2001) Skew-circulant preconditioners for systems of LMF-based ODE codes. Numer. Anal. Appl. LNCS. 93-101","DOI":"10.1007\/3-540-45262-1_12"},{"issue":"2","key":"2524_CR4","doi-asserted-by":"publisher","first-page":"221","DOI":"10.1080\/00029890.2006.11920300","volume":"113","author":"R Bhatia","year":"2006","unstructured":"Bhatia R (2006) Infinitely divisible matrices. Amer Math Monthly 113(2):221\u2013235","journal-title":"Amer Math Monthly"},{"issue":"2","key":"2524_CR5","doi-asserted-by":"publisher","first-page":"22","DOI":"10.1007\/s00283-010-9194-z","volume":"33","author":"R Bhatia","year":"2011","unstructured":"Bhatia R (2011) Min matrices and mean matrices. Math Intelligen 33(2):22\u201328","journal-title":"Math Intelligen"},{"issue":"1","key":"2524_CR6","doi-asserted-by":"publisher","first-page":"283","DOI":"10.1016\/j.cam.2005.08.047","volume":"200","author":"CM da Fonseca","year":"2007","unstructured":"da Fonseca CM (2007) On the eigenvalues of some tridiagonal matrices. J Comput Appl Math 200(1):283\u2013286","journal-title":"J Comput Appl Math"},{"issue":"3","key":"2524_CR7","first-page":"284","volume":"10","author":"O Deveci","year":"2016","unstructured":"Deveci O (2016) The Pell-circulant sequences and their applications. Maejo Int J Sci Technol 10(3):284\u2013293","journal-title":"Maejo Int J Sci Technol"},{"key":"2524_CR8","first-page":"107","volume":"108","author":"O Deveci","year":"2018","unstructured":"Deveci O (2018) On The Fibonacci-Circulant $$p$$-Sequences. Util. Math. 108:107\u2013124","journal-title":"Util. Math."},{"issue":"4","key":"2524_CR9","doi-asserted-by":"publisher","first-page":"1033","DOI":"10.1007\/s40995-017-0317-7","volume":"41","author":"O Deveci","year":"2017","unstructured":"Deveci O, Karaduman E, Campbell CM (2017) The Fibonacci-circulant sequences and their applications. Iran J Sci Technol Trans Sci 41(4):1033\u20131038","journal-title":"Iran J Sci Technol Trans Sci"},{"issue":"4","key":"2524_CR10","doi-asserted-by":"publisher","first-page":"378","DOI":"10.1137\/0106026","volume":"6","author":"WL Frank","year":"1958","unstructured":"Frank WL (1958) Computing eigenvalues of complex matrices by determinant evaluation and by methods of Danilewski and Wielandt. J Soc Ind Appl Math 6(4):378\u2013392","journal-title":"J Soc Ind Appl Math"},{"key":"2524_CR11","doi-asserted-by":"publisher","first-page":"375","DOI":"10.1007\/BF02240202","volume":"35","author":"J-F Hake","year":"1985","unstructured":"Hake J-F (1985) A remark on Frank matrices. Computing 35:375\u2013379","journal-title":"Computing"},{"key":"2524_CR12","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1090\/psapm\/040\/1059485","volume":"40","author":"RA Horn","year":"1990","unstructured":"Horn RA (1990) The Hadamard product. Proc Sympos Appl Math 40:87\u2013169","journal-title":"Proc Sympos Appl Math"},{"key":"2524_CR13","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511840371","volume-title":"Topics in matrix analysis","author":"RA Horn","year":"1991","unstructured":"Horn RA, Johnson CR (1991) Topics in matrix analysis. Cambridge University Press, Cambridge"},{"key":"2524_CR14","unstructured":"Jiang ZL, Jiang ZW (2012) On the norms of RFPLR circulant matrices with the Fibonacci and Lucas numbers. SCET (Xi\u2019an): 385-388"},{"key":"2524_CR15","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1007\/BF02936555","volume":"18","author":"ZL Jiang","year":"2005","unstructured":"Jiang ZL, Xu ZB (2005) Efficient algorithm for finding the inverse and the group inverse of FLS $$r$$-circulant matrix. J Appl Math Comput 18:45\u201357","journal-title":"J Appl Math Comput"},{"issue":"4","key":"2524_CR16","doi-asserted-by":"publisher","first-page":"2010","DOI":"10.3906\/mat-1811-95","volume":"43","author":"E K\u0131l\u0131c","year":"2019","unstructured":"K\u0131l\u0131c E, Ar\u0131kan T (2019) Studying new generalizations of Max-Min matrices with a novel approach. Turk J Math 43(4):2010\u20132024","journal-title":"Turk J Math"},{"key":"2524_CR17","doi-asserted-by":"publisher","first-page":"4559","DOI":"10.1007\/s12190-022-01717-y","volume":"68","author":"C K\u0131z\u0131late\u015f","year":"2022","unstructured":"K\u0131z\u0131late\u015f C, Terzio\u011flu N (2022) On $$r$$-min and $$r$$ -max matrices. J Appl Math Comput 68:4559\u20134588","journal-title":"J Appl Math Comput"},{"key":"2524_CR18","doi-asserted-by":"publisher","first-page":"312","DOI":"10.1186\/s13660-016-1255-1","volume":"2016","author":"C K\u0131z\u0131late\u015f","year":"2016","unstructured":"K\u0131z\u0131late\u015f C, Tuglu N (2016) On the bounds for the spectral norms of geometric circulant matrices. J. Inequal. Appl. 2016:312","journal-title":"J. Inequal. Appl."},{"issue":"2","key":"2524_CR19","first-page":"555","volume":"31","author":"C K\u0131z\u0131late\u015f","year":"2018","unstructured":"K\u0131z\u0131late\u015f C, Tuglu N (2018) On the norms of geometric and symmetric geometric circulant matrices with the Tribonacci number. Gazi Univ. J. Sci. 31(2):555\u2013567","journal-title":"Gazi Univ. J. Sci."},{"key":"2524_CR20","first-page":"353","volume":"85","author":"EG Kocer","year":"2007","unstructured":"Kocer EG, Mansour T, Tuglu N (2007) Norms of circulant and semicirculant matrices with Horadam\u2019s numbers. Ars Comb. 85:353\u2013359","journal-title":"Ars Comb."},{"key":"2524_CR21","volume-title":"Fibonacci and lucas numbers with applications","author":"T Koshy","year":"2019","unstructured":"Koshy T (2019) Fibonacci and lucas numbers with applications. Wiley, New York, NY, USA"},{"issue":"245","key":"2524_CR22","doi-asserted-by":"publisher","first-page":"279","DOI":"10.1090\/S0025-5718-03-01534-5","volume":"73","author":"JN Lyness","year":"2004","unstructured":"Lyness JN, S\u00f8revik T (2004) Four-dimensional lattice rules generated by skew-circulant matrices. Math Comput 73(245):279\u2013295","journal-title":"Math Comput"},{"key":"2524_CR23","first-page":"101","volume":"4","author":"M Mattila","year":"2016","unstructured":"Mattila M, Haukkanen P (2016) Studying the various properties of min and max matrices-elementary vs. more advanced methods. Spec Matrix 4:101\u2013109","journal-title":"Spec Matrix"},{"issue":"7","key":"2524_CR24","doi-asserted-by":"publisher","first-page":"17229","DOI":"10.3934\/math.2023880","volume":"8","author":"E\u00d6 Mersin","year":"2023","unstructured":"Mersin E\u00d6 (2023) Sturm\u2019s theorem for Min matrices. AIMS Math 8(7):17229\u201317245","journal-title":"AIMS Math"},{"issue":"4","key":"2524_CR25","doi-asserted-by":"publisher","first-page":"1002","DOI":"10.15672\/hujms.773281","volume":"50","author":"E\u00d6 Mersin","year":"2021","unstructured":"Mersin E\u00d6, Bahsi M (2021) Sturm theorem for the generalized Frank matrix. Hacet J Math Stat 50(4):1002\u20131011","journal-title":"Hacet J Math Stat"},{"issue":"2","key":"2524_CR26","doi-asserted-by":"publisher","first-page":"170","DOI":"10.36753\/mathenot.672621","volume":"8","author":"E\u00d6 Mersin","year":"2020","unstructured":"Mersin E\u00d6, Bahsi M, Maden AD (2020) Some properties of generalized Frank matrices. Math Sci Appl E-Notes 8(2):170\u2013177","journal-title":"Math Sci Appl E-Notes"},{"key":"2524_CR27","doi-asserted-by":"publisher","first-page":"221","DOI":"10.1007\/s00362-008-0174-8","volume":"50","author":"H Neudecker","year":"2009","unstructured":"Neudecker H, Trenkler G, Liu S (2009) Problem section. Stat Pap 50:221\u2013223","journal-title":"Stat Pap"},{"issue":"53","key":"2524_CR28","doi-asserted-by":"publisher","first-page":"855","DOI":"10.46939\/J.Sci.Arts-20.4-a07","volume":"4","author":"D Ozgul","year":"2020","unstructured":"Ozgul D, Bahsi M (2020) Min matrices with hyper Lucas numbers. J Sci Arts 4(53):855\u2013864","journal-title":"J Sci Arts"},{"key":"2524_CR29","doi-asserted-by":"publisher","first-page":"47","DOI":"10.11648\/j.acm.20150402.13","volume":"4","author":"SHJ Petroudi","year":"2015","unstructured":"Petroudi SHJ, Pirouz B (2015) A particular matrix, Its inversion and some norms. Appl Comput Math 4:47\u201352","journal-title":"Appl Comput Math"},{"issue":"48","key":"2524_CR30","first-page":"575","volume":"3","author":"E Polatl\u0131","year":"2019","unstructured":"Polatl\u0131 E (2019) On the bounds for the spectral norms of $$r$$ -circulant matrices with a type of catalan triangle numbers. J Sci Art 3(48):575\u2013578","journal-title":"J Sci Art"},{"key":"2524_CR31","unstructured":"P\u00f3lya G, Szeg\u0151 G (1925) Aufgaben und Lehrs\u00e4tze aus der Analysis, Zweiter Band: Funktionentheorie, Nullstellen, Polynome, Determinanten, Zahlentheorie. Grundlagen Math. Wiss. Einzeld. 20, Springer, Berlin"},{"issue":"1","key":"2524_CR32","doi-asserted-by":"publisher","first-page":"135","DOI":"10.2989\/16073606.2015.1024185","volume":"39","author":"B Radicic","year":"2016","unstructured":"Radicic B (2016) On $$k$$-circulant matrices (with geometric sequence). Quaest Math 39(1):135\u2013144","journal-title":"Quaest Math"},{"issue":"8","key":"2524_CR33","doi-asserted-by":"publisher","first-page":"2517","DOI":"10.2298\/FIL1708517R","volume":"31","author":"B Radicic","year":"2017","unstructured":"Radicic B (2017) On $$k$$-circulant matrices with arithmetic sequence. Filomat. 31(8):2517\u20132525","journal-title":"Filomat."},{"key":"2524_CR34","first-page":"2891","volume":"216","author":"SQ Shen","year":"2010","unstructured":"Shen SQ, Cen JM (2010) On the bounds for the norms of $$r$$ -circulant matrices with Fibonacci and Lucas numbers. Appl Math Comput 216:2891\u20132897","journal-title":"Appl Math Comput"},{"issue":"1","key":"2524_CR35","first-page":"42","volume":"12","author":"N Shen","year":"2013","unstructured":"Shen N, Jiang ZL, Li J (2013) On explicit determinants of the RFMLR and RLMFL circulant matrices involving certain famous numbers. WSEAS Trans Math 12(1):42\u201353","journal-title":"WSEAS Trans Math"},{"key":"2524_CR36","doi-asserted-by":"publisher","first-page":"14","DOI":"10.1186\/s13660-017-1608-4","volume":"2018","author":"BJ Shi","year":"2018","unstructured":"Shi BJ (2018) The spectral norms of geometric circulant matrices with the generalized $$k$$-Horadam numbers. J Inequal Appl 2018:14","journal-title":"J Inequal Appl"},{"key":"2524_CR37","first-page":"1","volume":"2021","author":"BJ Shi","year":"2021","unstructured":"Shi BJ (2021) On the norms of RFMLR-circulant matrices with the exponential and trigonometric functions. J Math 2021:1\u20139","journal-title":"J Math"},{"key":"2524_CR38","first-page":"125","volume":"160","author":"S Solak","year":"2005","unstructured":"Solak S (2005) On the norms of circulant matrices with the Fibonacci and Lucas numbers. Appl Math Comput 160:125\u2013132","journal-title":"Appl Math Comput"},{"key":"2524_CR39","doi-asserted-by":"publisher","first-page":"253","DOI":"10.1186\/s13660-015-0778-1","volume":"2015","author":"N Tuglu","year":"2015","unstructured":"Tuglu N, K\u0131z\u0131late\u015f C (2015) On the norms of circulant and $$r$$-circulant matrices with the hyperharmonic Fibonacci numbers. J. Inequal. Appl. 2015:253","journal-title":"J. Inequal. Appl."},{"key":"2524_CR40","doi-asserted-by":"publisher","first-page":"297","DOI":"10.1186\/s13662-015-0635-z","volume":"2015","author":"N Tuglu","year":"2015","unstructured":"Tuglu N, K\u0131z\u0131late\u015f C, Kesim S (2015) On the harmonic and hyperharmonic Fibonacci numbers. Adv Differ Equ 2015:297","journal-title":"Adv Differ Equ"},{"issue":"3","key":"2524_CR41","doi-asserted-by":"publisher","first-page":"835","DOI":"10.1137\/0907056","volume":"7","author":"JM Varch","year":"1986","unstructured":"Varch JM (1986) A generalization of the Frank matrix. SIAM J Sci Stat Comput 7(3):835\u2013839","journal-title":"SIAM J Sci Stat Comput"},{"key":"2524_CR42","doi-asserted-by":"crossref","unstructured":"Xu TT, Jiang ZL (2014) Explicit determinants of the RFP$$r$$LrR circulant and RLP$$r$$F$$r$$L circulant matrices involving some famous numbers. Abstr Appl Anal Article ID: 647030","DOI":"10.1155\/2014\/273680"},{"key":"2524_CR43","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-5797-2","volume-title":"Matrix theory. Basic results and techniques","author":"F Zhang","year":"1999","unstructured":"Zhang F (1999) Matrix theory. Basic results and techniques. Springer, New York"},{"key":"2524_CR44","doi-asserted-by":"crossref","unstructured":"Zhao G (2009) The improved nonsingularity on the $$r$$ -circulant matrices in signal processing. Proc. Int. Conf. Comp. Tech. Deve. (Kota Kinabalu\/Malaysia). 564-567","DOI":"10.1109\/ICCTD.2009.218"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-023-02524-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-023-02524-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-023-02524-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,11]],"date-time":"2024-02-11T11:21:12Z","timestamp":1707650472000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-023-02524-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,12,12]]},"references-count":44,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2024,2]]}},"alternative-id":["2524"],"URL":"https:\/\/doi.org\/10.1007\/s40314-023-02524-2","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,12,12]]},"assertion":[{"value":"22 June 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 October 2023","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 November 2023","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 December 2023","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"}}],"article-number":"19"}}