Hadamard operations on rational functions. (English) Zbl 0543.10006

van der Poorten, Alfred J.

Groupe Étude Anal. Ultramétrique, 10e Année, 1982/83, No. 1, Exp. No. 4, 11 p. (1984).

Suppose \(\sum b_ hX^ h\), \(\sum c_ hX^ h\) are Taylor series of rational functions. It is fairly easy to prove that their Hadamard product \(\sum(b_ hc_ h)X^ h\) is again rational. A long-standing problem, studied by Pólya, Pisot and David Cantor amongst others is whether the Hadamard quotient \(\sum(c_ h/b_ h)X^ h\) is rational when it possibly can be; for example when \(c_ h/b_ h\in\mathbb Z\) for all \(h\). This paper gives a fairly clumsy but basically correct proof of the most general form of the Hadamard quotient theorem: if there is a finitely generated subring \(R\) of a field \(K\) of characteristic zero over which the given functions are defined with \(a'_ h\in R\) and \(c_ h=a'_ hb_ h\), all \(h\), then there is a sequence \((a_ h)\) with \(c_ h=a_ hb_ h\), all \(h\), and the \(\sum a_ hX^ h\) is rational. The author suggests that he has techniques for proving a further result (Theorem B); his allegations are false.
A more readable outline of the proof of Theorem A appears in the author’s paper in Bull. Aust. Math. Soc. 29, 109–117 (1984; Zbl 0519.10009).

Reviewer: Alf van der Poorten (Killara)

MSC:

11B37	Recurrences
13J05	Power series rings

Keywords:

power series; rationality; exponential polynomial; Taylor series of rational functions; Hadamard product; Hadamard quotient theorem

Citations:

Zbl 0519.10009