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

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Abstract

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

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

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.

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  1. Institute of Applied Mathematics, Russian Academy of Sciences, Russia

    V. I. Parusnikov

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  1. V. I. Parusnikov
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__________

Translated from Matematicheskie Zametki, vol. 78, no. 6, 2005, pp. 892–906.

Original Russian Text Copyright ©2005 by V. I. Parusnikov.

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Parusnikov, V.I. A Generalization of Pincherle's Theorem to k-Term Recursion Relations. Math Notes 78, 827–840 (2005). https://doi.org/10.1007/s11006-005-0188-7

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  • DOI: https://doi.org/10.1007/s11006-005-0188-7

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