Abstract.
Let K be a field of characteristic 0 and let p, q, G 0 , G 1 , P ∈K[x], deg P ⩾ 1. Further, let the sequence of polynomials (G n (x)) n=0 ∞ be defined by the second order linear recurring sequence
In this paper we give conditions under which the diophantine equation G n (x) = G m (P(x)) has at most exp(1018) many solutions (n, m) ε ℤ2, n, m ⩾ 0. The proof uses a very recent result on S-unit equations over fields of characteristic 0 due to Evertse, Schlickewei and Schmidt [14]. Under the same conditions we present also bounds for the cardinality of the set
In the last part we specialize our results to certain families of orthogonal polynomials.
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This work was supported by the Austrian Science Foundation FWF, grant S8307-MAT.
The second author was supported by the Hungarian National Foundation for Scientific Research Grants No 16741 and 38225.
Received June 5, 2001; in revised form February 26, 2002
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ID="a" Dedicated to Edmund Hlawka on the occasion of his 85th birthday
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Fuchs, C., Pethö, A. & Tichy, R. On the Diophantine Equation G n (x) = G m (P(x)). Monatsh. Math. 137, 173–196 (2002). https://doi.org/10.1007/s00605-002-0497-9
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DOI: https://doi.org/10.1007/s00605-002-0497-9