On the diophantine equation Gn(x) = Gm(P(x))

Clemens Fuchs*, Attila Petho, Robert F. Tichy

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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 languageEnglish
Pages (from-to)173-196
Number of pages24
JournalMonatshefte fur Mathematik
Volume137
Issue number3
DOIs
Publication statusPublished - Nov 2002

Keywords

  • Diophantine equations
  • Linear recurring sequences
  • Orthogonal polynomials
  • S-unit equations

ASJC Scopus subject areas

  • Mathematics(all)

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