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We consider the group $\mathrm{Aut}(T)$ of isometries of a semi-homogeneoustree $T=T_{q_+,q_-}$ with valencies $q_+ +1$ and $q_- +1$ and its two orbits$V_+$, $V_-$ respectively. We make use of the action of $\mathrm{Aut} (T)$ toequip the spaces of finitely supported radial functions on each of $V_\pm$ withconvolution products, hence with a notion of positive definite functions. The$\ell^1$-functions radial around a root vertex $v_0\in V_+$ form an abelianconvolution algebra. We study its multiplicative functionals, called sphericalfunctions, given by eigenfunctions of the nearest-neighbor isotropic transitionoperator (the Laplace operator on $T$, and determine which of them are positivedefinite. Each positive definite function gives rise to a unitaryrepresentation of $\mathrm{Aut}(T)$; in this way, we produce a series ofunitary spherical representations. For $q_+<q_-$, the representation whosespherical function has eigenvalue 0 is square-integrable.

Positive definite functions on semi-homogeneous trees and spherical representations