2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/71136In this paper, we will consider R-transform theory and R-transform calculus for compatible noncommutative probability space and amagamated noncommutative probability space. By doing this, we can realize the relation between scalar-valued R-transforms and operator-valued moment series, under the compatibility. Also, we can see that there is a big gap between freeness and amalgamated freeness.37 pagesOperator AlgebrasFunctional Analysis47-xxtext