Continued fractions and irrational functions
type: Item
instance of: scholarly article
title: Continued fractions and irrational functions
zbMATH Open document ID: 0714.30004
DOI: 10.1016/0377-0427(90)90365-7
author: R. M. Hovstad
published in: Journal of Computational and Applied Mathematics
publication date: 1990-01-01T00:00:00Z
review text: Denoting by ord \(p\geq 0\) the order of the zero point at \(z=0\) of a polynomial p(z), the following result is proved. In the continued fraction \(^{\infty}_{k=1}\frac{a_ k(z)}{b_ k(z)}\), where \(a_ k(z)\), \(b_ k(z)\) are polynomials with \(a_ k(0)=0\) and \(b_ k(0)\neq 0\) for \(k\geq 1\), we suppose that \[ \sum^{k+1}_{i=1}ord a_ i- \sum^{k}_{i=1}\max (\deg a_ i,\deg b_ i) \] is bounded below for k belonging to the infinite set of positive integers. Then this continued fraction corresponds to a rational function at \(z=0\) if and only if there exists a sequence \(\{p_ k(z)\}_{k\geq 0}\) of polynomials obeying \[ a_ k(z)p_{k-1}(z)=b_ k(z)p_ k(z)+p_{k+1}(z) \] for \(k\geq 1\) and \[ ord p_ k=\sum^{k}_{i=1}ord a_ i,\quad \deg p_ k=\sum^{k- 1}_{i=1}\max (\deg a_ i,\deg b_ i)+c \] for k sufficiently large, where c is a constant. Some corollaries of this result are also shown.
Mathematics Subject Classification ID: 30B70
zbMATH DE Number: 4175318
zbMATH Keywords: polynomial
reviewed by: Péter Kiss
MaRDI profile type: Publication
full work available at URL: 0377-0427(90)90365-7
OpenAlex ID: W2046454934
cites work: The Classical Irrationality Problem for T-Fractions
cites work: A simple continued fraction test for the irrationality of functions
cites work: Convergence of Continued Fractions
cites work: A general continued fraction expansion
cites work: Certain continued fractions associated with the Padé table
cites work: Q3241331
description: scientific article; zbMATH DE number 4175318