A note on the Lie complexity and beyond

In a recent paper, Jason P. Bell and Jeffrey Shallit introduced the notion of Lie complexity and proved that the Lie complexity function of an automatic sequence is automatic. In this note, we give more facts concerning Lie complexity and define the extended Lie complexity and the prefix Lie complexity. Further, we prove that some proprieties of Lie complexity also hold for the extended Lie complexity. Particularly, we prove that the extended Lie complexity function and the first-order difference sequence of the prefix Lie complexity function of an automatic sequence are both automatic. This article is dedicated to Jean-Paul Allouche on the occasion of his 70th birthday.