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Mixed periodic Jacobi continued fractions. (English) Zbl 0604.10028

We can associate with a Jacobi matrix a Jacobi continued fraction by \[ \phi (z)=\frac{b^ 2_ 0}{z-a_ 1-}\quad \frac{b_ 1^ 2}{z-a_ 2- }\quad \frac{b^ 2_ 2}{z-a_ 3-}...\quad. \] Under a suitable condition \(\phi\) (z) can be described in a Stieltjes transform \(\phi (z)=\int^{+\infty}_{-\infty}d\mu (x)/(z-x)\) for some Stieltjes measure \(d\mu\) (x) on the real axis. The paper investigates what kind of measures \(d\mu\) (x) give a mixed periodic Jacobi continued fraction. The author gives complete characterizations of these functions and concrete methods to construct them. The Stieltjes measure \(d\mu\) (x) are also explicitly described.
Reviewer: P.Kiss

MSC:

11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
11A55 Continued fractions
28A25 Integration with respect to measures and other set functions
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