Computing semi-commuting differential operators in one and multiple variables

We discuss the concept of what we refer to as semi-commuting linear differential operators. Such operators hold commuting operators as a special case. In particular, we discuss a heuristic by which one may construct such operators. Restricting to the case in which one such operator is of degree two, we construct families of linear differential operators semi-commuting with some named operators governing special functions (with a focus on the hypergeometric case, as it holds many other cases as reductions); operators commuting with such special degree two operators will necessarily be contained in these families. In the partial differential operator case, we consider examples in the form of the wave equation with a variable wave speed, and then hypergeometric operators on two variables (such operators define Appell functions). AMS subject classifications: 47E05, 13N10, 32W50, 47F05, 81Q15