This paper is dedicated to the two classical transcendental functions: The locus of points for which powers commute, and the locus of points for which powers associate. These classical functions however are considered in a new perspective: as holomorphic solutions of ODEs passing over the points of singularity of these ODEs.
Generally, solution functions which are holomorphic at singular points of the phase space of ODEs were studied in [2,3], and it was shown in [3], that certain holomorphic functions may satisfy only singular rational ODEs. This is the frame in which the function of commuting or associating powers are considered in this paper.
First we obtain several types of ODEs satisfied by these functions. The obtained ODEs happen to have singular points, yet the solutions are proved to be holomorphic at these points, and their Taylor expansions are obtained. However it is not yet known whether these two transcendental functions can satisfy a regular rational ODE at the respective special points. The article also poses an open question about remarkable inequalities related to the commuting powers.