%0 Journal Article %T Addition and multiplication of beta-expansions in generalized Tribonacci base %A Petr Ambro£¿ %A Zuzana Mas¨¢kov¨¢ %A Edita Pelantov¨¢ %J Discrete Mathematics & Theoretical Computer Science %D 2007 %I Discrete Mathematics & Theoretical Computer Science %X We study properties of ¦Â-numeration systems, where ¦Â > 1 is the real root of the polynomial x 3 - mx 2 - x - 1, m ¡Ê , m ¡Ý 1. We consider arithmetic operations on the set of ¦Â-integers, i.e., on the set of numbers whose greedy expansion in base ¦Â has no fractional part. We show that the number of fractional digits arising under addition of ¦Â-integers is at most 5 for m ¡Ý 3 and 6 for m = 2, whereas under multiplication it is at most 6 for all m ¡Ý 2. We thus generalize the results known for Tribonacci numeration system, i.e., for m = 1. We summarize the combinatorial properties of infinite words naturally defined by ¦Â-integers. We point out the differences between the structure of ¦Â-integers in cases m = 1 and m ¡Ý 2. %U http://www.dmtcs.org/dmtcs-ojs/index.php/dmtcs/article/view/650