Taylor's theorem and functional equations

Abstract

Several authors (cf. [1]—[10]) studied functional equations stemming from Mean Value Theorems. In the present paper we solve an equation that originates from Taylor's formula, viz.¶¶\( f(y) = \sum\limits_{k=0}^{n-2} \gamma_k(x)(y-x)^k + \phi (( 1-\frac{1}{n}) x + \frac{1}{n}y)(y-x)^{n-1} \).¶No regularity assumptions are assumed and all the functions are unknown, defined and taking values in Abelian groups.