combinatorial arguments about generalized stirling numbers

Studying expressions of the form (f(x)d)^p, where d=d/dx is the derivative operator, goes back to scherk’s ph.d. thesis in 1823. the special case f(x)=x^k in his work was also discussed by lang at the end of twentieth century. more generalizations were also considered by blasiak et al. here we discuss about some algebraic and combinatorial aspects of such problems