Euler, R.; Gallardo, L. H. On explicit formulae and linear recurrent sequences. (English) Zbl 1255.11061 Acta Math. Univ. Comen., New Ser. 80, No. 2, 213-219 (2011). 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. Reviewer: Štefan Porubský (Praha) Cited in 1 ReviewCited in 1 Document MSC: 11T55 Arithmetic theory of polynomial rings over finite fields 11B39 Fibonacci and Lucas numbers and polynomials and generalizations 11T06 Polynomials over finite fields Keywords:polynomials; Euclidean division; finite fields; even characteristic Citations:Zbl 0223.13007; Zbl 0252.13002; Zbl 1122.11008; Zbl 1123.05300 PDFBibTeX XMLCite \textit{R. Euler} and \textit{L. H. Gallardo}, Acta Math. Univ. Comen., New Ser. 80, No. 2, 213--219 (2011; Zbl 1255.11061)