Continued fractions and irrational functions
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.
scientific article; zbMATH DE number 4175318