Convergence properties of linear recurrence sequences

This paper provides a survey of the dissertation of the rst named author 6]. The thesis deals with recurrence sequences fu n g 1 n=0 of complex numbers satisfying (1) a k (n)u n+k + a k?1 (n)u n+k?1 This kind of sequences plays an important role in analysis (the theory of orthogonal poly-nomials) and in combinatorics. Important applications in number theory can be found in Ap ery's proof of the irrationality of (3) = P 1 n=1 n ?3 and in other derivations of ir-rationality measures (cf. G.V. Chudnovsky 5] p. 344.) In most applications k = 2 and the coeecients a 2 ; a 1 ; a 0 are polynomials. We shall deal with the asymptotic behaviour of sequences fu n g as n ! 1; in particular the existence of lim n!1 u n+1 =u n : At the end we shall give some applications, one of which concerns the solution of a problem posed by Perron. It will appear that there are obvious similarities with the theory of linear diieren-tial equations, but also notable diierences. The second author thanks several participants of the conference for their helpful comments.