Some facts that should be better known, especially about rational functions.

*(English)*Zbl 0687.10007
Number theory and applications, Proc. NATO ASI, Banff/Can. 1988, NATO ASI Ser., Ser. C 265, 497-528 (1989).

[For the entire collection see Zbl 0676.00005.]

The present paper gives an overview on subjects that are intimately connected with Taylor expansions of rational functions, recurrence sequences and their growth rate, exponential polynomials, recognition of rational functions, Hadamard quotients and capacity theory. The author has taken a very active part in the development of this circle of ideas which has grown very rapidly in the last ten or fifteen years. It is a welcome opportunity to have a unified and lively written account of this progress. To give a complete description of all topics discussed would exceed the bounds of this review. Suffice it to say that this paper is a valuable reference for workers in recurrence sequences and related fields, and an entertaining story for all those interested in the funny behaviour of coefficients of Taylor expansions of rational functions.

The present paper gives an overview on subjects that are intimately connected with Taylor expansions of rational functions, recurrence sequences and their growth rate, exponential polynomials, recognition of rational functions, Hadamard quotients and capacity theory. The author has taken a very active part in the development of this circle of ideas which has grown very rapidly in the last ten or fifteen years. It is a welcome opportunity to have a unified and lively written account of this progress. To give a complete description of all topics discussed would exceed the bounds of this review. Suffice it to say that this paper is a valuable reference for workers in recurrence sequences and related fields, and an entertaining story for all those interested in the funny behaviour of coefficients of Taylor expansions of rational functions.

Reviewer: F.Beukers

##### MSC:

11B37 | Recurrences |

30B10 | Power series (including lacunary series) in one complex variable |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

41A58 | Series expansions (e.g., Taylor, Lidstone series, but not Fourier series) |