Automatic sequences and parity of partition functions

Abstract

Let $k\ge 2$ be an integer, ℓ a prime and $F_{\ell }$ the finite field with ℓ elements. A sequence ${\left(a\right(n\left)\right)}_{n\ge 0}$ is called k-automatic if there exists a deterministic finite automaton with output that reads the canonical base-k representation of n and the outputs $a\left(n\right)$. We apply the properties of automatic sequences to prove the transcendence of a formal power series over $F_{\ell }\left(X\right)$ related to infinite products. As applications, the parity results of various partition functions are obtained, including the root partition function and the prime parts partition function. We also establish the transcendence of the power series associated with holomorphic modular forms with integer coefficients.