q-recursive sequences and their asymptotic analysis

Daniel Krenn
Paris Lodron University of Salzburg

Date(s) : 07/11/2022   iCal
15 h 00 min - 16 h 00 min

In this talk, we consider Stern’s diatomic sequence, the number of non-zero elements in some generalized Pascal’s triangle and the number of unbordered factors in the Thue-Morse sequence as running examples. All these sequences can be defined recursively and lead to the concept of so-called qq-recursive sequences. Here qq is an integer and at least 22, and qq-recursive sequences are sequences which satisfy a specific type of recurrence relation: Roughly speaking, every subsequence whose indices run through a residue class modulo qMq^M is a linear combination of subsequences where for each of these subsequences, the indices run through a residue class modulo qmq^m for some m<Mm < M.

It turns out that this property is quite natural and many combinatorial sequences are in fact qq-recursive. We will see that qq-recursive sequences are related to qq-regular sequences and a qq-linear representation of a sequence can be computed easily. Our main focus is the asymptotic behavior of the summatory functions of qq-recursive sequences. Beside general results, we present a precise asymptotic analysis of our three examples. For the first two sequences, our analysis even leads to precise formulae without error terms.

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More info: https://www.i2m.univ-amu.fr/wiki/Combinatorics-on-Words-seminar/

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