Almansi functions near infinity in R n M.Damlakhi, M.Al-Qurashi and V.Anandam Abstract: A characterization of continuous functions of the form u(x)= m  i=0 |x| i h i (x), where each h i (x) is a harmonic function defined outside a compact set in R n , n ≥ 2, is obtained by means of a generalized version of the Kelvin transformation. Mathematics Subject Classification (2000): 31B30. Key words : Almansi representation, Kelvin transformation. 1 Introduction A C 4 -function u(x) on |x| <r in R n ,n ≥ 2, is biharmonic ( that is ∆ 2 u =0) if and only if u is of the form u(x)= |x| 2 u 1 (x)+ u 0 (x) where u 1 (x) and u 0 (x) are harmonic functions on |x| < r. This representation is very useful in obtaining certain properties of biharmonic functions, particularly the growth properties at infinity similar to those of harmonic functions. (Nicolesco [6,pp.13- 21] details some of these properties for polyharmonic functions.) But such a representation for a biharmonic function is possible only on a star domain with center 0. For example, the biharmonic function u(x)= log |x| on |x| > 1 in R 4 does not have a representation of the form u(x)= |x| 2 u 1 (x)+ u 0 (x) with u 1 and u 0 being harmonic on |x| > 1. In this note, we identify the continuous functions of the form m ∑ i=0 |x| i h i (x) where h i are harmonic functions defined outside a compact set in R n , n ≥ 2, by means of a generalized version of the Kelvin transformation in R n and study their relation to polyharmonic functions defined by the equation ∆ m u =0. Also, for such functions, the Liouville-Picard theorem, the Laurent decomposition the- orem, the limit at infinity, the property of analyticity etc. are investigated. 1