Mathematical Notes

, Volume 78, Issue 5–6, pp 827–840 | Cite as

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

  • V. I. Parusnikov


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.

Key words

Pincherle theorem recursion relation multidimensional continued fraction 


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Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • V. I. Parusnikov
    • 1
  1. 1.Institute of Applied MathematicsRussian Academy of SciencesRussia

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