Expansion Series of $f(x)=x^x$ And Characterization of its Coefficients

Abstract: In this paper we study the development in Taylor series of the function $f(x)=x^x$. First section establishes a recursive relationship among successive derivatives of the function by using the coefficients defined therein. From recursion between the derivatives you get one general description of them (section 2). Finally, section 3 has the main result, the expansion series. Section 4 deals with the coefficients: characterization, their relationship with rencontre numbers and their emergence as coefficients in certain polynomial families. The specific use of some of these polynomials allows eventually go deeper in the description of the series.