Discrete ”orders of infinity”. (English) Zbl 0602.26003

Boshernitzan, Michael

Am. J. Math. 106, 1147-1198 (1984).

The author gives the discrete analogue of his theory of orders of infinity [J. Anal. Math. 39, 235-255 (1981; Zbl 0539.26002); ibid. 41, 130-167 (1982; Zbl 0539.26003); Am. J. Math. 106, 1067-1089 (1984; preceding review)].
The fundamental concepts are the rings (we omit the author’s subscript S) B of real sequences (those which have ultimately equal terms are considered identical), \(\Delta\) B-rings (rings of sequences in B which are closed under translations \(a_ n\mapsto a_{n+k}\)) and ordered B- and \(\Delta\) B-rings and fields (in which the terms of sequences are ultimately sign-preserving). E is the intersection of all maximal ordered \(\Delta\) B-fields.
The paper and the theory is extensive. It contains among others results on the growth properties of sequences which satisfy recurrences of the form \(a_{n+1}=b_ na_ n+c_ na_{n-1}\) \((\{b_ n\}\) and \(\{c_ n\}\) in an ordered \(\Delta\) B-field) and culminates in results like the following. Let \(\{p_ n\}\) and \(\{q_ n\}\) be in an ordered \(\Delta\) B- field and suppose that \(\lim_{n\to \infty}(p_ n/q_ n)=\alpha\) exists. Suppose also that \(\{q_ n\alpha -p_ n\}\) is bounded from both sides by polynomial sequences in n. Then \(\alpha\) is rational if and only if also \(\{q_ n\alpha -p_ n\}\) is itself a polynomial sequence. This is applied to finding necessary and sufficient conditions for the rationality of continued fractions. Several open problems and conjectures are stated.

Reviewer: J.Aczél

MSC:

26A12	Rate of growth of functions, orders of infinity, slowly varying functions
41A25	Rate of convergence, degree of approximation
12H05	Differential algebra

Keywords:

maximal discrete Hardy fields; discrete analogue; orders of infinity; rings of sequences; recurrences; rationality of continued fractions

Citations:

Zbl 0602.26002; Zbl 0539.26002; Zbl 0539.26003