In this paper we explicitly determine the Macdonald formula for spherical functions on any locally finite, regular and affine Bruhat-Tits building, by constructing the finite difference equations that must be satisfied and explaining how they arise, by only using the geometric properties of the building.

Let p,q be positive integers. The groups $U_p\left(ℂ\right)$ and $U_p\left(ℂ\right)\times U_q\left(ℂ\right)$ act on the Heisenberg group $H_{p,q}:=M_{p,q}\left(ℂ\right)\times ℝ$ canonically as groups of automorphisms, where $M_{p,q}\left(ℂ\right)$ is the vector space of all complex p × q matrices. The associated orbit spaces may be identified with ${\Pi }_q\times ℝ$ and ${\Xi }_q\times ℝ$ respectively, ${\Pi }_q$ being the cone of positive semidefinite matrices and ${\Xi }_q$ the Weyl chamber $x\in {ℝ}^q:x₁\ge \cdots \ge x_q\ge 0$. In this paper we compute the associated convolutions on ${\Pi }_q\times ℝ$ and ${\Xi }_q\times ℝ$ explicitly, depending on p. Moreover, we extend these convolutions by analytic continuation to series of convolution...