Abstract
There are many identities including Fibonacci numbers. However, few determinants of \(n\times n\) matrix which is equal to the Fibonacci number have been known. In 1974, Proskuryakov showed the first such an example in his Linear Algebra book, though it is believed that the first person is Lucas. Nevertheless, in 1875, Glaisher gave several determinants of matrices which are equal to the Bernoulli, Euler, Cauchy and more numbers. By studying Cameron’s operator in terms of determinants, we introduce the technique to produce many examples of \(n\times n\) matrix which is equal to the Fibonacci-like numbers.
Similar content being viewed by others
References
Ballot, C.: On functions expressible as words on a pair of Beatty sequences. J. Integer Seq. 20, 24 (2017)
Brioschi, F.: Sulle funzioni Bernoulliane ed Euleriane. Ann. Mat. Pura Appl. 1, 260–263 (1858)
Cahill, N.D., D’Errico, J.R., Narayan, D.A., Narayan, J.Y.: Fibonacci determinants. Coll. Math. J. 33, 221–225 (2002)
Cahill, N.D., D’Errico, J.R., Spence, J.P.: Complex factorizations of Fibonacci and Lucas numbers. Fibonacci Q. 41, 13–19 (2003)
Cameron, P.J.: Some sequences of integers. Discrete Math. 75, 89–102 (1989)
Charalambides, C.A.: Enumerative Combinatorics (Discrete Mathematics and its Applications). CRC Press, Boca Raton (2002)
Donaghey, R., Shapiro, L.W.: Motzkin numbers. J. Comb. Theory Ser. A 23, 291–301 (1977)
Dykema, K.J.: Multilinear function series and transforms in free probability theory. Adv. Math. 208, 351–407 (2007)
Fan, C.K.: Structure of a Hecke algebra quotient. J. Am. Math. Soc. 10, 139–167 (1997)
Glaisher, J.W.L.: Expressions for Laplace’s coefficients, Bernoullian and Eulerian numbers etc. as determinants. Messenger 6, 49–63 (1875)
Hermes, J.: Anzahl der Zerlegungen einer ganzen rationalen Zahl in Summanden. Math. Ann. 45, 371–380 (1894)
Komatsu, T.: Poly-Cauchy numbers. Kyushu J. Math. 67, 143–153 (2013)
Komatsu, T., Ramirez, J.L.: Some determinants involving incomplete Fubini numbers. An. Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat. 26(3), 143–170 (2018)
Komatsu, T., Yuan, P.: Hypergeometric Cauchy numbers and polynomials. Acta Math. Hungar. 153, 382–400 (2017)
Koshy, T.: Fibonacci and Lucas Numbers with Applications. Wiley, New York (2001)
Koshy, T.: Catalan Numbers with Applications. Oxford University Press, New York (2009)
Muir, T.: The Theory of Determinants in the Historical Order of Development, vol. 4. Dover Publications, New York (1960)
Proskuryakov, I.V.: Problems in linear algebra, Translated from the Russian by George Yankovsky. Revised from the 1974 Russian edition. “Mir”, Moscow, 453 pp (1978)
Ramírez, J.L., Sirvent, V.F.: A note on the \(k\)-Narayana sequence. Ann. Math. Inform. 45, 91–105 (2015)
Sloane, N.J.A.: The on-line encyclopedia of integer sequences. http://oeis.org. (2020)
Strang, G.: Linear Algebra and its Applications. Brooks/Cole, Pacific Grove (1988)
Trojovský, P.: On a sequence of tridiagonal matrices whose determinants are Fibonacci numbers. Int. J. Pure Appl. Math. 102, 527–532 (2015)
Trudi, N.: Intorno ad alcune formole di sviluppo, Rendic. dell’ Accad. Napoli, 135–143 (1862)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Komatsu, T. Fibonacci determinants with Cameron’s operator. Bol. Soc. Mat. Mex. 26, 841–863 (2020). https://doi.org/10.1007/s40590-020-00286-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40590-020-00286-z