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

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

### Citations:

Zbl 0602.26002; Zbl 0539.26002; Zbl 0539.26003
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