Let be an algebraically closed field of characteristic 0,
. Let further
and the sequence of
be defined by the recursion
for all .
In this talk we are given a survey on results concerning the equation
in integers with , where
. We assume
that are transcendental over , but algebraically dependent,
holds. Our journey is based on joint works with Cl. Fuchs
and R.F. Tichy as well as on a paper of U. Zannier. If
then (1) has up to two exceptional families only finitely
many solutions and Zannier proved a quite good bound for the number of
solutions. In the general case we have also a quite satisfactory
description of the exceptional cases.
We finish our talk by some open problem.
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Please send comments and corrections to Thomas Klausner.