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

Clemens Fuchs*, Attila Petho, Robert F. Tichy

*Korrespondierende/r Autor/-in für diese Arbeit

Publikation: Beitrag in einer FachzeitschriftArtikelBegutachtung

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.

Originalspracheenglisch
Seiten (von - bis)173-196
Seitenumfang24
FachzeitschriftMonatshefte fur Mathematik
Jahrgang137
Ausgabenummer3
DOIs
PublikationsstatusVeröffentlicht - Nov. 2002

ASJC Scopus subject areas

  • Mathematik (insg.)

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