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Linear recurrence sequences and periodicity of multidimensional continued fractions

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Abstract

Multidimensional continued fractions generalize classical continued fractions with the aim of providing periodic representations of algebraic irrationalities by means of integer sequences. We provide a characterization for periodicity of Jacobi–Perron algorithm by means of linear recurrence sequences. In particular, we prove that partial quotients of a multidimensional continued fraction are periodic if and only if numerators and denominators of convergents are linear recurrence sequences, generalizing similar results that hold for classical continued fractions.

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Acknowledgments

I would like to thank Prof. Umberto Cerruti for introducing me to these beautiful problems. Thanks to the anonymous referee whose comments improved the readability and presentation of the paper.

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  1. Department of Mathematics, University of Turin, Via Carlo Alberto 10, 10123, Turin, Italy

    Nadir Murru

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  1. Nadir Murru
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Correspondence to Nadir Murru.

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Murru, N. Linear recurrence sequences and periodicity of multidimensional continued fractions. Ramanujan J 44, 115–124 (2017). https://doi.org/10.1007/s11139-016-9820-2

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  • DOI: https://doi.org/10.1007/s11139-016-9820-2

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