Transcendence for Pisot Morphic Words over an Algebraic Base

Abstract

It is known that for a uniform morphic sequence $\boldsymbol u = \langle u_n\rangle_{n=0}^\infty$ and an algebraic number $\beta$ such that $|\beta|>1$, the number $[\![\boldsymbol{u} ]\!]_\beta:=\sum_{n=0}^\infty \frac{u_n}{\beta^n}$ either lies in $\mathbb Q(\beta)$ or is transcendental. In this paper we show a similar rational-transcendental dichotomy for sequences defined by irreducible Pisot morphisms. Subject to the Pisot conjecture (an irreducible Pisot morphism has pure discrete spectrum), we generalise the latter result to arbitrary finite alphabets. In certain cases we are able to show transcendence of $[\![\boldsymbol{u}]\!]_{\beta}$ outright. In particular, for $k\geq 2$, if $\boldsymbol u$ is the $k$-bonacci word then $[\![\boldsymbol{u}]\!]_{\beta}$ is transcendental.