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2008 | OriginalPaper | Chapter
The Set of Solutions of Some Equation for Linear Recurrence Sequences
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In [SS1] Schlickewei and Schmidt studied the solutions of various linear equations involving members of recurrence sequences. Most of them are of the form
(A)
$$ F_1 \left( {x_1 } \right) + \cdots + F_n \left( {x_n } \right) = 0 $$
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 α
ji
(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 α
j0
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., α
ji
/α
jh
is not a root of unity for
h ≠
i.