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Jordan forms and \(n\)th order linear recurrences. (English) Zbl 1352.11026

Summary: Let \(p\) be a prime number with \(p\neq 2\). We consider sequences generated by \(n\)th order linear recurrence relations over the finite field \(\mathbb Z_p\). In the first part of this paper we generalize some of the ideas in our paper [Missouri J. Math. Sci. 25, No. 1, 27–36 (2013; Zbl 1292.11037)] to \(n\)th order linear recurrences. We then consider the case where the characteristic polynomial of the recurrence has one root in \(\mathbb Z_p\) of multiplicity \(n\). In this case, we show that the corresponding recurrence can be generated by a relatively simple matrix.

MSC:

11B50 Sequences (mod \(m\))
11B39 Fibonacci and Lucas numbers and polynomials and generalizations

Citations:

Zbl 1292.11037

References:

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