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

TL;DR

This paper develops the Taylor series expansion of the function x^x, deriving recursive relationships for derivatives, characterizing coefficients, and linking them to rencontre numbers and polynomial families.

Contribution

It introduces a recursive method for derivatives of x^x and characterizes the series coefficients, connecting them to combinatorial numbers and polynomial structures.

Findings

Derived recursive relationships for derivatives of x^x

Characterized series coefficients and linked them to rencontre numbers

Connected coefficients to specific polynomial families

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.