The Generalization of the Pinkerle Theorem for the K-term Recurrence Equations.

Abstract: In 1894 Pinkerle proved the theorem, which assist the connection between the existence of the so-called minimal solutions of a system of recurrence relations and the convergence of related continued fraction. In this paper we consider solutions of the infinite system of the (k+1)-term recurrence relations qn=j=1∑k-1pk-j,nqn-j, p1,n≠ 0, n = 0,1, … , with coefficients p in a field F. The connection between such recurrence relations and (k-2)-dimensional continued fractions is stated. The analogy of the Pinkerle theorem is proved.Note: Research direction:Mathematical problems and theory of numerical method