Geodesic
A Generalization of Pincherle's Theorem to $k$-Term Recursion Relations
Matematičeskie zametki, Tome 78 (2005) no. 6, pp. 892-906.

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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. 
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V. I. Parusnikov. A Generalization of Pincherle's Theorem to $k$-Term Recursion Relations. Matematičeskie zametki, Tome 78 (2005) no. 6, pp. 892-906. http://geodesic.mathdoc.fr/item/MZM_2005_78_6_a8/

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