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Generalization of the Fibonacci Sequence to n Dimensions

Published online by Cambridge University Press:  20 November 2018

George N. Raney
George N. Raney*
Affiliation:
University of Connecticut, Storrs, Connecticut
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We introduce certain n X n matrices with integral elements that constitute a free semigroup with identity and generate the n-dimensional unimodular group. In terms of these matrices we define a certain sequence of n-dimensional vectors, which we show is the natural generalization to n dimensions of the Fibonacci sequence. Connections between the generalized Fibonacci sequences and certain Jacobi polynomials are found. The various basic identities concerning the Fibonacci numbers are shown to have natural extensions to n dimensions, and in some cases the proofs are rendered quite brief by the use of known theorems on matrices.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 332 - 349
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Appel, K. I. and Djorup, F. M., On the group generated by a free semigroup, Proc. Amer. Math. Soc, 15 (1964), 838–840.Google Scholar
2. Artin, E., Geometric algebra (New York, 1957), p. 163.Google Scholar
3. Bernstein, L., Periodische Kettenbrüche beliebiger Periodenlänge, Math. Z., 86 (1964), 128–135.Google Scholar
4. Birkhoff, G., Uniformly semi-primitive multiplicative processes, Trans. Amer. Math. Soc, 104 (1962), 37–51.Google Scholar
5. Coxeter, H. S. M., Introduction tc geometry (New York, 1961). p. 168.Google Scholar
6. Litoff, O., On the commutator subgroup of the general linear group, Proc. Amer. Math. Soc., 6 (1955), 465–470.Google Scholar
7. Lucas, E., Théorie des fonctions numériques simplement périodiques, Amer. J. Math., 1 (1878), 184–240, 289-321.Google Scholar
8. Szegö, G., Orthogonal polynomials (Providence, 1939). p. 28.Google Scholar