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 $$q_n = sumlimits_{j - 1}^{k - 1} {_{Pk - j,n} } q_{n - j} ,quad p_{1,n} ne 0,quad n = 0,1, ldots ,$$ 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.