eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-06-18
27:1
27:18
10.4230/LIPIcs.AofA.2018.27
article
Analysis of Summatory Functions of Regular Sequences: Transducer and Pascal's Rhombus
Heuberger, Clemens
1
https://orcid.org/0000-0003-0082-7334
Krenn, Daniel
1
https://orcid.org/0000-0001-8076-8535
Prodinger, Helmut
2
https://orcid.org/0000-0002-0009-8015
Institut für Mathematik, Alpen-Adria-Universität Klagenfurt, Universitätsstraße 65--67, 9020 Klagenfurt am Wörthersee, Austria
Department of Mathematical Sciences, Stellenbosch University, 7602 Stellenbosch, South Africa
The summatory function of a q-regular sequence in the sense of Allouche and Shallit is analysed asymptotically. The result is a sum of periodic fluctuations multiplied by a scaling factor. Each summand corresponds to an eigenvalues of absolute value larger than the joint spectral radius of the matrices of a linear representation of the sequence. The Fourier coefficients of the fluctuations are expressed in terms of residues of the corresponding Dirichlet generating function. A known pseudo Tauberian argument is extended in order to overcome convergence problems in Mellin-Perron summation.
Two examples are discussed in more detail: The case of sequences defined as the sum of outputs written by a transducer when reading a q-ary expansion of the input and the number of odd entries in the rows of Pascal's rhombus.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol110-aofa2018/LIPIcs.AofA.2018.27/LIPIcs.AofA.2018.27.pdf
Regular sequence
Mellin-Perron summation
summatory function
transducer
Pascal's rhombus