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The Set of Solutions of Some Equation for Linear Recurrence Sequences

  • Viktor Losert
Part of the Developments in Mathematics book series (DEVM, volume 16)

Abstract

In [SS1] Schlickewei and Schmidt studied the solutions of various linear equations involving members of recurrence sequences. Most of them are of the form
$$ F_1 \left( {x_1 } \right) + \cdots + F_n \left( {x_n } \right) = 0 $$
(A)
with x i ∈ ℤ, where \( F_j \left( x \right) = \sum\nolimits_{i = 0}^{r_j } {f_{ji} \left( x \right)\alpha _{ji}^x \left( {j = 1, \ldots ,n} \right)} \), r j > 0 with given polynomials f ji and nonzero numbers αji (thus for each j, (F j (x))x∈ℤ is a linear recurrence sequence, see also [ST, Sec.C]). The general assumption of [SS1, p.220] is that αj0 is a root of unity and that f ji ≠ 0 for i > 0 (f j0 may be zero), j = 1, ..., n. Furthermore, they restrict to nondegenerate sequences, i.e., αjijh is not a root of unity for h ≠ i.

Keywords

Linear recurrence sequences polynomial-exponential type 

2000 Mathematics subject classification

11D61 11B37 

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References

  1. [La]
    Laurent, M.: Équations exponentielles-polynômes et suites récurrentes linéaires. II. J. Number Theory 31, 24–53 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
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    Losert, V.: Two equations for linear recurrence sequences. Acta Arith. 119, 109–147 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  3. [SS1]
    Schlickewei, H.R, Schmidt, W. M.: Linear equations in members of recurrence sequences. Ann. Sc. Norm. Super. Pisa Cl. Sci. IV Ser. 20, 219–246 (1993)zbMATHMathSciNetGoogle Scholar
  4. [SS2]
    Schlickewei, H.R, Schmidt, W.M.: The number of solutions of polynomial-exponential equations. Compos. Math. 120, 193–225 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  5. [ST]
    Shorey, T.N., Tijdeman, R.: Exponential Diophantine Equations. Cambridge University Press, Cambridge (1986)Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Viktor Losert
    • 1
  1. 1.Institut für MathematikUniversität WienWienAustria

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