Convergence conditions for vector Stieltjes continued fractions (Q1599255)

This nicely written paper gives the full proofs of two results on the convergence of \(p\)-dimensional continued fractions previously announced in Russian in [Dokl. Russian Acad. Sci. 375, No. 6 (2000)]. The proofs are based on the spectral properties of a two banded matrix, connected with a recurrence relation of order \(p+1\) with first and last coefficient different from zero and the intermediate coefficients all zero. The vector generalization of the continued fraction is based on the Jacobi-Perron algorithm where product and quotient of two vectors \(a, b\) from \(\mathbb{C}^p\) are defined by \[ a\cdot b=(a_1b_1,\ldots,a_pb_p),\qquad \left({1\over a}\right) = \left({1\over a_p},{a_1\over a_p},\ldots,{a_{p-1}\over a_p}\right). \]