Automatic sequences defined by Theta functions and some infinite products

2019/07/17 by Shuo Li, Li, Shuo
Computer Science · Mathematics · #Combinatorics (math.CO) #Computability, Logic, AI Algorithms #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1907.07572

openalex publication_date 2019/07/17 · openalex created_date 2019/07/23 · openalex updated_date 2026/07/28

Abstract

Let p(x) ∈ C(x) be a rational function satisfying the condition p(0)=1 and q an integer larger than 1, in this article we will consider the power expansion of the infinite product f(x)=∏_s=0^∞f(x^q^s)=∑_i=0^∞c_ixⁱ, and study when the sequence (c_i)_{i ∈ N} is q-automatic. The main result is that for given integers q ≥ 2 and d ≥ 0, there exist finitely many polynomials of degree d defined over the field of rational numbers Q, such that ∏_s=0^∞f(x^q^s)=∑_i=1^∞c_ixⁱ is a q-automatic power series.

Related

• A certain reciprocal power sum is never an integer
• Badly approximable infinite products of quadratic polynomials
• Certain infinite products in terms of MacMahon type series
• ϕ-Thue-Morse sequences and infinite products
• On the Collection of Integers that Index the Fixed Points of Maps on the Space of Rational Functions