# Stochastic Analysis Common-Iddo Ben-Ari (University of Connecticut)-A Probabilistic Approach to Generalized Zeckendorf Decompositions

Mathematics - Lecture/Discussion

Monday, October 7, 2013

4:00 PM-5:00 PM

Stratton Hall

202

ABSTRACT: Zeckendorf's Theorem states that every positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers, if we start the sequence 1, 2, 3, 5,... This result has been generalized to decompositions arising from other recurrence relations, and to properties of the decompositions, most notably, Lekkerkerker's Theorem which gives the mean number of summands. The theorem was originally proved using continued fraction techniques, but recently a more combinatorial approach has had great success in attacking this and related problems, such as the distribution between gaps of summands. We introduce a unified probabilistic framework and show how this machinery allows to reprove and generalize all existing results and obtain new results. The main idea is that the digits appearing in the decomposition are obtained by a simple change of measure for some Markov chain.

Suggested Audiences:
Adult, College

E-mail:
ma-chair@wpi.edu

Last Modified: October 3, 2013 at 10:55 AM

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## Stochastic Analysis Common-Iddo Ben-Ari (University of Connecticut)-A Probabilistic Approach to Generalized Zeckendorf Decompositions

Mathematics - Lecture/Discussion

Monday, October 7, 2013

4:00 PM-5:00 PM

Stratton Hall

202

ABSTRACT: Zeckendorf's Theorem states that every positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers, if we start the sequence 1, 2, 3, 5,... This result has been generalized to decompositions arising from other recurrence relations, and to properties of the decompositions, most notably, Lekkerkerker's Theorem which gives the mean number of summands. The theorem was originally proved using continued fraction techniques, but recently a more combinatorial approach has had great success in attacking this and related problems, such as the distribution between gaps of summands. We introduce a unified probabilistic framework and show how this machinery allows to reprove and generalize all existing results and obtain new results. The main idea is that the digits appearing in the decomposition are obtained by a simple change of measure for some Markov chain.

Suggested Audiences: Adult, College

E-mail: ma-chair@wpi.edu

Last Modified: October 3, 2013 at 10:55 AM

Powered by the Social Web - Bringing people together through Events, Places, & Common Interests