## 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 G_{n+2}(x) = p(x)G_{n+1}(x) + q(x)G_{n}(x), for n ≥ 0. In this paper we give conditions under which the diophantine equation G_{n}(x)= G_{m}(P(x)) has at most exp(10^{18}) many solutions (n, m) ∈ ℤ^{2}, 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 G_{n} (x) = c G_{m}(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

- Mathematics(all)