A Generalization of Pincherle's Theorem to $k$-Term Recursion Relations

In 1894, Pincherle proved a theorem relating the existence of a minimal solution of three-term recursion relations to the convergence of a continued fraction. The present paper deals with solutions of an infinite system
$$q_n=\sum_{j=1}^{k-1}p_{k-j,n}q_{n-j}, \qquad p_{1,n}\ne0, \quad n=0,1,\dots,$$
of $k$-term recursion relations with coefficients in a field $F$. We study the connection between such relations and multidimensional ($(k-2)$-dimensional) continued fractions. A multidimensional analog of Pincherle's theorem is established.