Abstract
Let K be a field of characteristic 0 and let p, q, G0, G1, P ∈ K[x], deg P ≥ 1. Further, let the sequence of polynomials (Gn(x))n=0∞ be defined by the second order linear recurring sequence Gn+2(x) = p(x)Gn+1(x) + q(x)Gn(x), for n ≥ 0. In this paper we give conditions under which the diophantine equation Gn(x)= Gm(P(x)) has at most exp(1018) many solutions (n, m) ∈ 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. Under the same conditions we present also bounds for the cardinality of the set {(m,n) ∈ ℕ | m ≠ n, ∃ c ∈ K\{0} such that Gn (x) = c Gm(P(x))}. In the last part we specialize our results to certain families of orthogonal polynomials.
Original language | English |
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Pages (from-to) | 173-196 |
Number of pages | 24 |
Journal | Monatshefte fur Mathematik |
Volume | 137 |
Issue number | 3 |
DOIs | |
Publication status | Published - Nov 2002 |
Keywords
- Diophantine equations
- Linear recurring sequences
- Orthogonal polynomials
- S-unit equations
ASJC Scopus subject areas
- General Mathematics