##
**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.

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 |