Asymptotics of Mahler recurrences: The cyclotomic case
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.
scientific article; zbMATH DE number 912496
scientific article