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

2014/10/17 by Oliver Planes, Planes, Oliver
Mathematics · #11B75 (Secondary) #41A58 (Primary) #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #Differential Equations and Numerical Methods #FOS: Mathematics #math.CA #msc:11B75 #msc:41A58

paper · pdf · doi:10.48550/arxiv.1411.1712

14 pages

arxiv created 2014/10/17 · openalex publication_date 2014/10/17 · arxiv updated 2014/11/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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