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Development of an Individual-Tree Basal Area Increment Model using a Linear Mixed-Effects Approach
04:35

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Published on: July 3, 2020

Multivariate linear recurrences and power series division.

Herwig Hauser1, Christoph Koutschan

  • 1Fakultät für Mathematik, Universität Wien, Nordbergstraße 15, A-1090 Wien, Austria ; Institut für Mathematik, Technikerstraße 13, Universität Innsbruck, A-6020, Austria.

Discrete Mathematics
|March 14, 2013
PubMed
Summary

This study reinterprets generating functions for multivariate linear recurrences using formal power series division theorems. This approach offers clearer structural insights and simpler proofs for both constant and polynomial coefficient cases.

Keywords:
Formal power seriesLinear recurrence equationMultivariate sequencePerfect operatorPower series division[Formula: see text]-finite recurrence

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Area of Science:

  • Combinatorics
  • Algebraic Combinatorics
  • Formal Power Series

Background:

  • Multivariate linear recurrences with constant coefficients are fundamental in combinatorics.
  • Generating functions provide a powerful tool for analyzing these recurrences.
  • Existing methods for analyzing these functions can be complex.

Purpose of the Study:

  • To provide a novel reinterpretation of existing results on generating functions for multivariate linear recurrences.
  • To offer simplified and conceptual proofs for these results.
  • To extend the methodology to handle recurrences with polynomial coefficients.

Main Methods:

  • Utilizing division theorems for formal power series.
  • Applying these theorems to reinterpret the work of Bousquet-Mélou and Petkovšek.
  • Extending the division concept to differential operators.

Main Results:

  • A structural clarification of generating functions for multivariate linear recurrences.
  • Concise and conceptual proofs derived from the division theorems.
  • An analogous method for analyzing recurrences with polynomial coefficients.

Conclusions:

  • The division theorems for formal power series offer a unifying framework for studying linear recurrences.
  • This approach simplifies the analysis and extends its applicability.
  • The findings contribute to a deeper understanding of combinatorial structures and generating functions.