Partial slice regularity and Fueter's Theorem in several quaternionic variables

We extend some definitions and give new results about the theory of slice analysis in several quaternionic variables. The sets of slice functions which are respectively slice, slice regular and circular w.r.t. given variables are characterized. We introduce new notions of partial spherical value and derivative for functions of several variables that extend those of one variable. We recover some of their properties as circularity, harmonicity, some relations with differential operators and a Leibniz rule w.r.t. the slice product as well as studying their behavior in the context of several variables. Then, we prove our main result, which is a generalization of Fueter's Theorem for slice regular functions in several variables. This extends the link between slice regular and axially monogenic functions well known in the one variable context.