Fibonacci determinants with Cameron’s operator

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.

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Correspondence to Takao Komatsu.

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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

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Keywords

  • Operator
  • Fibonacci numbers
  • Restricted numbers
  • Associated numbers
  • Determinants

Mathematics Subject Classification

  • Primary 11B39
  • Secondary 05A15
  • 05A19
  • 15B05
  • 11B68
  • 11B73
  • 11B75
  • 11C20