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On explicit formulae and linear recurrent sequences. (English) Zbl 1255.11061

Let \(R\) be a commutative ring with \(1\) and \(f(X)\in R[X]\) a monic polynomial of degree \(n\). S. Agou [C. R. Acad. Sci., Paris, Sér. A 273, 209–211 (1971; Zbl 0223.13007)] and [Publ. Dépt. Math., Lyon 8, 107–121 (1971; Zbl 0252.13002)] found explicit formulae for the quotient and remainder when \(X^m\) is divided by \(f(X)\) for \(m\geq n\). In the present paper the authors continue Agou’s work in the case when \(R={\mathbb F}_q[X]\) and \(q\) is even and show that there is a class of Fibonacci-like linear recurrences over the ring of polynomials \({\mathbb F}_{2^k}[X]\) with infinitely many vanishing terms. They also notice that the main results by H. Belbachir and F. Bencherif [Integers 6, A12, 17 p. (2006; Zbl 1122.11008)] extending those by J. McLaughlin [Integers 4, A19, 15 p. (2004; Zbl 1123.05300)] are already obtained in Agou’s second paper mentioned above.

MSC:

11T55 Arithmetic theory of polynomial rings over finite fields
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11T06 Polynomials over finite fields
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