Polyregularity of the dot product of slice regular functions

In this paper, we are concerned with the S-polyregularity the regular dot product of slice regular functions. We prove that the product of a slice regular function and right quaternionic polynomial function is a S-polyregular function and we determinate its exact order. The general case of the product of any two slice regular functions is also discussed. In fact, we provide sufficient and necessary conditions to the product of slice regular functions be a S-polyregular function. The obtained results are then extended to the product of S-polyregular functions and remain valid for a special dot product. As consequences we obtain linearization theorems for such S-polyregular products with respect to the slice regular functions.