Asymptotics of Mahler recurrences: The cyclotomic case
type: Item
instance of: scholarly article
title: Asymptotics of Mahler recurrences: The cyclotomic case
zbMATH Open document ID: 0869.11080
author: Philippe Flajolet
author: Philippe Dumas
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 1996-11-24T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/228754
full work available at URL: jtnb8-1.html
review text: Mahler studied the transcendence properties of functions such as \(\sum z^{B^n}\) and \(\prod (1-z^{B^n})^{-1}\) and, more generally, solutions of functional equations of the shape \(c_0(z) f(z)+c_1(z) f(z^B)+ \cdots +c_N(z) f(z^{B^N})=0\). He developed new methods to resolve questions about the transcendence and algebraic independence of functions of this type and their values at algebraic points. The Mahler functions appear in many applications, including the theory of partitions and the theory of automata and as generating functions for various combinatorial problems. De Bruijn gave very precise asymptotic estimates for binary partitions and partitions whose parts are powers of \(B\) by analysing the particular Mahler function \(\prod (1-z^{B^n})^{-1}\). The analytic technique is based on Mellin transforms and the circle method. In this paper, the method is developed to obtain the asymptotic expansion \[ \prod^\infty_{k=0} {1\over\Phi_a(z^{B^k})} \sim\exp \left[{\log^2 \rho \over 2\log B} +(1+ \kappa) \log\rho +n\rho+ {1\over 2} \log n\rho \right] \times \sum^\infty_{k=0} {1\over (n \rho)^{k/2}} \omega_k \left({\log\rho \over\varphi (a)\log B} \right), \] where \(\Phi_a\) is the cyclotomic polynomial of order \(a\), \(\rho\) is defined by \(\log\rho \sim- \log n+ \log\log n-\log \log 2+ o(1)\) and the functions \(\omega_k\) are analytic and have period 1. The waves which appear in the expansion are typical and arise in various problems about restricted partitions [see \textit{K. F. Roth} and \textit{G. Szekeres}, Q. J. Math., Oxf. II. Ser. 5, 241-259 (1954; Zbl 0057.03902)] which gives the asymptotic behaviour of the number of partitions of \(n\) into \(m\) parts and also in the spectral study of automata [see \textit{E. Bombieri} and \textit{J. E. Taylor}, J. Phys. 47, Colloq. C3, Suppl. au No. 7, 19-28 (1986; Zbl 0693.52002)] which discusses the same phenomenon in the context of the properties of 1-dimensional quasicrystals.
Mathematics Subject Classification ID: 11P82
Mathematics Subject Classification ID: 11B85
Mathematics Subject Classification ID: 11B37
zbMATH DE Number: 912496
zbMATH Keywords: Mahler functions
zbMATH Keywords: partitions
zbMATH Keywords: automata
zbMATH Keywords: generating functions
zbMATH Keywords: Mellin transforms
zbMATH Keywords: circle method
zbMATH Keywords: asymptotic expansion
zbMATH Keywords: cyclotomic polynomial
reviewed by: John H. Loxton
MaRDI profile type: Publication
OpenAlex ID: W2332126946
cites work: Q3785142
cites work: The ring of \(k\)-regular sequences
cites work: Q3801620
cites work: Introduction to algorithms
cites work: Q5786798
cites work: Q3250680
cites work: Sur la fonction sommatoire de la fonction 'somme des chiffres'
cites work: Q5767841
cites work: Exact asymptotics of divide-and-conquer recurrences
cites work: Mellin transforms and asymptotics. The mergesort recurrence
cites work: Mellin transforms and asymptotics: Digital sums
cites work: Periods in strings
cites work: Arithmetic properties of lacunary power series with integral coefficients
cites work: Fifty years as a Mathematician
cites work: Enumeration of bordered words, le langage de la vache-qui-rit
cites work: Power and Exponential Sums of Digital Sums Related to Binomial Coefficient Parity
cites work: Divide and Conquer Heuristics for Minimum Weighted Euclidean Matching
cites work: Q3735790
cites work: Q3993278
DOI: 10.5802/JTNB.154
Recommended article: A NOTE ON AN ASYMPTOTIC VERSION OF A PROBLEM OF MAHLER
Recommended article: Q3377523
Recommended article: The distribution of Mahler's measures of reciprocal polynomials
Recommended article: Asymptotics of some convolutional recurrences
Recommended article: Higher Mahler measure for cyclotomic polynomials and Lehmer's question
Recommended article: Asymptotic Results for Class Number Divisibility in Cyclotomic Fields
Recommended article: Q5666649
Recommended article: DEGREE-ONE MAHLER FUNCTIONS: ASYMPTOTICS, APPLICATIONS AND SPECULATIONS
Recommended article: Asymptotic behaviour to solutions to linear recurrences and sequences of Möbius-transformations
Recommended article: Q5666650
description: scientific article; zbMATH DE number 912496