For each complex number z, we construct an operator H defined on the space z of all complex-valued functions on a homogeneous tree. This operator has the Ž property that if a function f is harmonic i.e., the local averaging operator fixes the . Ž . Ž values of f , then H f is z-harmonic i.e., the local averaging operator multiplies z Ž .
. H f by z . Because the Laplacian is the local averaging operator minus the z identity, a z-harmonic function is an eigenfunction of the Laplacian relative to the eigenvalue z y 1. We show that all z-harmonic functions are in the image of H , z and we compare H to another well-known operator which converts harmonic z functions to z-harmonic functions. We then study the problem of representing a function as the integral of z-harmonic functions with respect to a distribution. In particular, if a function grows no faster than exponentially, then the distribution is Ž . Ž . Ž . of the form f z,y d z , where f z,y is a z-harmonic function and is the Lebesgue measure in .ރ If the base of the growth is sufficiently small, the w Ž . Ž . x distribution is supported over the interval y2 q r q q 1 , 2 q r q q 1 . ᮊ 1998 ' '