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.
@InProceedings{heuberger_et_al:LIPIcs.AofA.2018.27, author = {Heuberger, Clemens and Krenn, Daniel and Prodinger, Helmut}, title = {{Analysis of Summatory Functions of Regular Sequences: Transducer and Pascal's Rhombus}}, booktitle = {29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018)}, pages = {27:1--27:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-078-1}, ISSN = {1868-8969}, year = {2018}, volume = {110}, editor = {Fill, James Allen and Ward, Mark Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2018.27}, URN = {urn:nbn:de:0030-drops-89204}, doi = {10.4230/LIPIcs.AofA.2018.27}, annote = {Keywords: Regular sequence, Mellin-Perron summation, summatory function, transducer, Pascal's rhombus} }
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